This quest starts immediately after the quest Alduin's Bane is complete. However, it will be put on hold shortly after it starts for Season Unending if the Civil War questline is incomplete or not started.
After I defeated Alduin at the top of the Throat of the World, he fled from me. One of his dragon allies may be able to tell me where he's gone. With the help of the Jarl of Whiterun, I've trapped Odahviing in Dragonsreach. Odahviing revealed that Alduin's lair is located in the ancient temple of Skuldafn, which is only accessible by air. Odahviing has offered to take me there, in return for his freedom.
After you have concluded the peace treaty negotiation in Season Unending, you will learn that names of all dragons are actually Words of Power, and that Dragon Shouts can be used to call them. You will then receive the Words of Power for Odahviing, in order to lure him into the trap as negotiated with the Jarl of Whiterun. If you did not negotiate a treaty, these words can be learned from Paarthurnax at The Throat of the World or Esbern at Sky Haven Temple.
You can now spring the trap for Odahviing. Travel to Dragonsreach and confirm with the Jarl that you are ready. Exit through the doors behind the throne room, to the "Great Porch" (Esbern calls it, incorrectly, the "Courtyard"). From this large open area you can call Odahviing, using first his Dragon Shout, and then Dragonrend to draw him out of the sky. (Note: You may need to go up on the small balcony to the right and shout from there. When shouting, hold the button down for longer to complete the Call Dragon Shout.) Do not bother fighting him or trying to drain his power, as he is at this point invincible. Lure him into the enclosed area, and the trap will come down on his neck and hold him in place. He will then speak to the Dragonborn, and offer to take them to Skuldafn in exchange for his freedom.
Note: If your followers or NPCs insist on tanking Odahviing and keeping far away from the trap, attack him until he is "KO'd" and then retreat to the doors. When Odahviing's HP is restored (since he is not killable at this point), he should attempt to follow you and get trapped.
At this point, Farengar may come in and start running tests on Odahviing, with amusing results. When you choose to free Odahviing, a guard on the second floor will help you. Speak with Odahviing again, and he'll carry you to Alduin's portal to Sovngarde via a cutscene.
When you release the captured dragon, he will attack you and you cannot talk to him. Walk out the door and wait for 1 hour and walk back in. All of the NPC's should be calmed down and you should be able to talk to him.
PC Dragonrend doesn't activate the encounter with the dragon and instead circles above. Hitting the dragon while he passes extremely close usually triggers the event. Re-loading the quest, from moment before entering the hall, also helps. OR Another way to fix this problem is to exit Dragonsreach, then wait for 20 hours and reenter.
If you're experiencing bugs listed below, typing "setstage mq301 200" into the console when Odahviing is trapped will remove his hostility and let you talk to him.
If Odahviing doesn't speak to you once he's been trapped, it's because the Jarl wasn't at the Porch before you called the dragon. If Vignar Grey-Mane becomes Jarl as a result of the Stormcloak questline, then he's likely sitting in Jorrvaskr 24/7. You must do the first Companions mission (Take Up Arms) before Vignar will leave the building. Once you get back to the Porch, Vignar should be waiting for you and the mission can proceed.
If Odahviing doesn't speak to you once he's been trapped and you have completed the first Companions mission, simple reload the autosave and recapture Odahviing, he should now offer to talk.
Odahviing will run after you but will randomly stop halfway and cease to move and attack like there is an invisible wall where he can't move, attack, speak or etc.
If you attempt to save the first attacked guard by force shouting him back into the relatively safe trap area, Odahviing will swoop through the stone walls, the guard will be lifted and appear to be moonwalking. If he's close enough to the second floor walkway, you can inspect his corpse like regular, although he will still appear to be moonwalking in midair.
If Farengar is killed before Odahviing is captured, conversations will still continue between the Jarl, Odahviing, and Irileth as if he is performing tests on the dragon.
If you fast travel from Dragonsreach before the dragon walks onto the porch, he may disappear from the game entirely with the game prompting you to go out the dungeon, but there will not be a dragon in sight. Similarly, if you walk through the door and back out too early, he may be stuck half way in the ground. You can still talk to him as normal. Did not test flying.
360 PC There is a bug where the Jarl of Whiterun doesn't return to Dragonsreach, instead standing outside and the bottom of the stairs. You cannot tell him that you are ready for the trap to be set. Whenever you talk to him, he says, "I'll have my men ready as soon as I return to Whiterun". This can be fixed by replaying from a previous save. On the PC, this can be fixed with the console command moveto player after selecting the current Jarl.
360 If the Frostbreath Shout is used on him while he's flying, he may fall to ground below the porch. This is an instanced area he will fall outside of the map and the quest will be uncompletable.
When using a sneak attack that "one hits" elder dragons on him, it may bypass his health completely killing him and making the quest uncompletable.
When calling him after the quest, he has the option to be talked to, yet no dialogue will appear.
360 If the Dovahkiin runs back to the edge of the balcony as the trap falls, Odahviing may appear to trapped with his tail where his head would be. Fire will still shoot out when the mage tries his expirements and dialogue will still occur as if you were talking to his face. This is fixed after Odahviing is released.
If you use the wrong shout, your shout will not regenerate. Just reload your last save point and you should be fine.
During the battle with Odahviing, he may land on top of the roof and remain there without attacking. He won't move at all and nothing happens when his health is brought to zero.
PS3 Once you have Odahviing captured, if you make the guard you are supposed to talk to free Odahviing hostile, then make him go outside, you can attack him and make him fall to one knee. Then when you get arrested, once you leave jail or pay the fine, the guard will vanish and the quest marker will point to the Whiterun dungeon, but he is not there. Also, at this point there is a chance that when you release your follower, he/she will be hostile and try to kill you.
PS3 The guard you need to speak to about freeing Odahviing can be killed when Odahviing attacks after being called. This can simply be ignored by pulling the lever next to him.
If you have a bounty in Whiterun and you have bribed a guard to ignore it, when you try to set Odahviing free by talking to the guard, the guard will only say 'Wait, I know you' rendering the quest uncompletable. This can simply be ignored by pulling the lever next to him.
Occasionally, Odahviing will not take any damage from either the Dragonborn, any followers, or even any Guards that assist in the fight. This can be fixed by simply leading Odahviing into the castle which activates the trap thus continuing the quest.
There is a bug when you go to the trap, and when you use the "Call Dragon" shout Odahviing will not show up.