There are many various quests in Titan Quest. If you see an NPC with exclamation mark above him/her head, then you can activate a quest, if you speak with NPC.

There are 2 types of quests:

Main Quests: These missions are about the game story. You must complete these quest to complete the game.

Side Quests: These quests are optional missions. There are many of them in each Act. You have to kill unique monsters, speak with NPCs, deliver messages..etc. in these quest.

Act 1 QuestsEdit

Act 2 QuestsEdit

Act 3 QuestsEdit

Act 4 QuestsEdit

Act 5 QuestsEdit

List of Side Quest

Act I - Greece:

 1. Monstrous Brigands
Just as you walk out of Helos a shepherd will come running and tell you that
his friend has been dragged into a cave by satyrs. Continue with quest 2 of
the main story and stop along to way to enter a cave hidden on the left side
of the road. Enter it, kill the satyrs swarming around and at the back of the
cave you will find Tellis, the hostage. He will thank you and give you your
Location: Helos, Helos Village
Reward: Random item, 300 XP

2. The Cornered Man
On your way to talk to the Spartan general, just after entering the Woods area,
follow the trail of monsters on the right side of the road and it will lead to a
lumber camp set on fire by satyrs. Kill all of them and the surrounded man will
thank you for saving his life. (Although he is practically ruined now :P)
Location: Helos, Laconia Woods
Reward: 1500 gold, 300 XP

3. Medicines Waylaid
On the way to the Spartan War Camp you will encounter a messenger from a village
plagued by a strange illness. He has been sent to bring medicine, but his way
back is now blocked by satyrs led by some sort of champion, he says. Proceed
through the pass, kill the 3 or 4 monsters who "welcome" you and you should see
the aura you should be accustomed to by now surrounding a big, nasty monster.
Prepare some potions, dispose of everything you can around him before he sees
you, but afterwards concentrate only on bringing him down. Once you succeed,
return to the village emissary.
Location: Helos, Laconia Hills
Reward: Replenish Life and Energy, some potions, 350 XP

4. The Lost Dowry
          Just before the bridge you need to cross in order to get to the Spartan camp,
          a Spartan peasant will approach you and ask you to retrieve the dowry for his
          daughter's wedding, which has been stolen by satyrs after the beasts had raided
          his cart. Enter the nearby cave fight your way to the back (why can't the stuff
          we are looking for be RIGHT at the beginning of some cave or dungeon? Grrr...),
          open the chest to get some valuable loot (or if not, at least a bit of cash) and
          on the right side of the cave kill the big monster surrounded by the well-known
          aura. He will drop the quest item you need. Return to the peasant for your 
          Location: Sparta, Laconia Hills, just before the bridge
          Reward: Relic Shard, 400 XP
          5. Ancient of War
          Just as you exit the Spartan War Camp, you will find Euthycles, a soldier
          concerned about his mentor's fate. He is somewhere in the Pellana Valley. I
          suggest solving this quest after you have killed Nessus. After leaving his camp,
          go forward and right and you should find Hippias, the "ancient of war", sitting
          in a circle of beast corpses. He apparently decided he should rest for a while,
          after he had slain several monsters. Return to Euthycles in the camp, who will
          be happy to hear his mentor still hasn't lost his touch.
          Location: Sparta, Spartan War Camp
          Reward: 2000 gold, 500XP
          6. The Poisoned Spring
          A young citizen from Tegea will inform you that the water has been infected and
          now his sister and neighbour are suffering from a terrible disease. He suspects
          foul play so he sends you to search for the cause of this infection in the 
          Tegean Woods. Take the northwest exit from Tegea, fight your way through arachnids and meet
          Pegaea the water naiad. She will direct you to the Cave behind her. Clear the
          entire cave of beasts (be sure to empty it, kill everything moving around in 
          that cave), open all chests, get the loot and exit. Talk to Pegaea to complete
          the quest.
          Location: Tegea
          Reward: Health Points (+75 in my case), 1500 XP
          7. Skeleton Raiders
          Just as you enter Megara, a citizen named Nireus will tell you that the people
          are terrified because undead beings have been ravaging the area. You are to
          find the cause and restore peace to Megara. Proceed further into the city and
          talk to Autolycus, who will also reveal the cause: three undead princes, who
          have found shelter in an abandoned and collapsed temple outside Megara. So,
          prepare some potions and take the path leading out of the city. In the far 
          north and after hefty battles you will find the temple. Enter it and search for
          the three princes. Luckily you don't have to face them together, since they are
          placed in different places in this large crypt. I can't give any specific 
          directions since they seem to spawn automatically. What I can tell you though is 
          that at least one of them will be in a room with an ice turret device. After you 
          have disposed of them (and ransacked the whole area; do not miss the small 
          sarcophagus-like crypts) return to Autolycus.
          Location: Megara
          Reward: loot, 2500 XP
          8. News of a Shipwreck
          Near the shipyard in Megara, you can talk to Anton, a shipwrecked sailor who
          wants to know what happened to the rest of the crew and the arms and gold (ding!
          sounds interesting...) they were transporting. This quest is solved right after
          prevailing in the fight with Polyphemus the Cyclops (see main quest 3). Start
          walking along the shore, near the rocks. About 12 drowned soldiers will spawn
          out of the ground. Dispose of them and open the chests around them (see, it was
          worth it).
          Location: Megara
          Reward: loot (almost 10000 gold + great items), 2000 XP
          9. The Good Centaur
          Northeast of the Bridge into Ambrossos from Boetia you will encounter Chiron,
          a noble centaur. He doesn't understand why the creatures of the woods have
          been enraged and asks you to retrieve his magical bow, stolen by Ino, a vile
          princess who was occupied Chiron's grove. So head into the cave, dispose of
          the creatures around and you should see Ino in front of you, surrounded by the
          aura. She will however run every time you see her. Corner her down to the back
          of the cave and kill her using melee (after you killed her minions) or ranged
          attacks, but in the case you've got to be careful since her ranged weapon can
          do a lot of damage. Anyway, after you sent her to a much nicer place, take the
          bow and any other loot and return to Chiron for your great reward.
          Location: Ambrossos
          Reward: 2 ATP (attribute points), 3000 XP
          10. Goods Abandoned
Just near the exit of the village in Ambrossos, near the merchant, there is
another trader, who doesn't have anything to sell however. He was ambushed
by an undead army and he lost all his goods. He is happy to have survived so
he considers this event as a fair deal and tells you that whoever finds his
goods can have them. Great. Stroll to the first rebirth fountain you find (near
the olive farmer) and head west from there. Descend to the most western part of
the map and kill the several skeletons that spawn in that area (not too many
though, I was expecting a horde). Anyway, open all the chests to finish the
Location: Ambrossos
Reward: loot, 1000 XP

11. A Proper Offering
Ok...This quest starts in Megara, but you have to get to Ambrossos to solve it.
Keep going on, solving side quests along the way, until you reach the olive
farmer. Go west and you should see the olive tree, guarded by something BIG.
It's actually Arachne, the spider queen. This is one tough battle, you should
know. She summons all sorts of Spiders and she also creates a nasty fog that
drains your health FAST. However, run if necessary, because she doesn't
regenerate life. When she is dead, click on the pedestal and pick up the
branch it will drop. Continue with the main quest "The Words of The Oracle" 
and give the branch to the temple attendant to finish this lengthy side quest.
Location: Megara
Reward: loot, + 5% permanent resistance to one type of damage

Note: You don't have to solve this quest to get to see the Oracle

12. A Master Blacksmith
Admetos in Ambrossos will tell you of a remarkable blacksmith who lives alone,
somewhere on the road to Delphi. He is actually in Lower Delphi just before
the bridge to Crisaeos Falls. Approach to him to end this mini quest, but don't 
rush to buy anything since the merchants in Delphi are far better stocked up.
Location: Ambrossos
Reward: 1500 XP

13. The Grieving Widow
Iodame can be found in the plaza near the portal in Delphi. She is haunted by
dreams of her late husband, who can not find his rest until the vile creature
that killed him is sent to hell. The horrible creature is part boar, part man.
The widow will ask you to kill the monster. You will find it in a remote
compartment of the Parnassus Caves. Defeating this monster is really a joke,
since he only takes 4 or 5 hits to go down and doesn't deal a lot of damage.
Once you disposed of him return to Delphi via the portal stone and talk to
Iodame to end this quest
Location: Delphi
Reward: 3000 XP

14. Trapped in the Ruins
Near the entrance to the Monster Encampment, you should find a Spartan patrol.
They captured a fierce monster, but now they are stuck with him, as they can not
kill him (one of their men has been badly hurt... I wonder how they caught him!)
Anyway, they ask you to kill the beast. While the game will still list the 
Cyclops as the toughest creature you've killed, this creature is far more 
dangerous. So if you aren't at least around level 14 you should skip this quest.
The creature you are to kill is a magical limos who can deal almost 350 damage
in melee, thus probably draining more than half of your life bar. He also has
a ranged poisonous, life draining attack. Your best bet for this fight is to
hit him with some spells or ranged attack, have a pet harassing him and keep
charging. You should eventually defeat him, although it might take a few tries.
Location: Kephisos Valley
Reward: magical weapon, 3000 XP

15. Spartans Lost
Isarchus can be found next to the Spartan captain who will initiate the quest
listed above. He will tell you that another patrol has gone missing and they
haven't heard of them. Cross the bridge to the Monster Encampment and head
east through the swamp. To the far east, you should find four confused Spartans.
Talk to their leader, return to the Kephisos Valley and talk to Isarchus to
end the quest.
Location: Kephisos Valley
Reward: 4000 XP