Below in chronological order by series are issues when membership changes occur (SPOILERS!):

The Avengers (1963)

• The Avengers #2 (Nov. 1963) - The Hulk leaves the team after realising the other Avengers don't trust him.
• The Avengers #4 (Mar. 1964) - Captain America joins The Avengers!
• The Avengers #9 (Oct. 1964) - Wonder Man is sent undercover by The Masters of Evil to infiltrate The Avengers and joins the team when he "rescues" them from Baron Zemo and his team.
• The Avengers #10 (Nov. 1964) - Wonder Man betrays the Masters of Evil and gives his life to save The Avengers.
• The Avengers #16 (May 1965) - The first major change in lineup as founding members Thor, Iron Man, Giant Man and The Wasp all leave, while Hawkeye, Quicksilver and The Scarlet Witch are recruited as replacements. This issue is where the concept "Older Order Changeth" was coined by writer/editor Stan Lee.
• The Avengers #19 (Aug. 1965) - Swordsman breaks into Avengers Mansion and proves he deserves a place in the team by besting The Avengers in one on one combat.
• The Avengers #20 (Sep. 1965) - Swordsman's stint in the team proves to be short lived as he turns out to be an assassin working for The Mandarin.
• The Avengers #28 (May 1966) - Hank Pym returns as Goliath and The Wasp also rejoins. In this issue the duo also reveal their secret identities to the rest of the team.
• The Avengers #45 (Oct. 1967) - Hercules is made an official member after fighting alongside The Avengers against The Mandarin.
• The Avengers #47 (Dec. 1967) – Captain America quits to create a life for Steve Rogers.
• The Avengers #49 (Feb. 1968) – Quicksilver and The Scarlet Witch quit to rejoin Magneto's Brotherhood of Evil Mutants.
• The Avengers #52 (May 1968) - The Black Panther rescues The Avengers from death at the hands of The Grim Reaper and joins the team.
• The Avengers #58 (Nov. 1968) - The Vision proves his worth by battling Captain America, Iron Man and Thor. Also having previously rescued the other Avengers from Ultron, he is given membership.
• The Avengers #71 (Dec. 1969) - The Black Knight (Dane Whitman) rescues The Avengers from death at the hands of Kang The Conqueror and is rewarded with membership of Earth's Mightiest Heroes. Strangely Black Knight doesn't appear in the near future issues of The Avengers as a recurring character. It is assumed he has taken up reserve membership and still operates in England.
• The Avengers #75 (Apr. 1970) - Yellow Jacket (Hank Pym) and The Wasp take a sabbatical from The Avengers so Hank can take part in a government research project in Alaska. This is the second time the couple have left The Avengers. Also in this issue, Quicksilver and Scarlet Witch rejoin the team.
• The Avengers #79 (Aug. 1970) - The Vision feels out of place being an android in a team full of humans and quits the team.
• The Avengers #80 (Sep. 1970) - The Vision's self induced exile does't last long as he returns to The Avengers to help his new ally Red Wolf.
• The Avengers #91 (Aug. 1971) - Hank Pym (as Yellow Jacket) feels he is no longer powerful enough to be an Avenger and his time will be best spent in the lab thus formerly resigns. The Wasp decides to leave with her husband and follows suit.
• The Avengers #92 (Sep 1971) - Thor, Iron Man and Captain America accuse Goliath (Clint Barton), Quicksilver, The Scarlet Witch and The Vision of tarnishing The Avengers name by harbouring the fugitive Kree alien Captain Marvel. Based on this the founding members dismiss the others and disassemble the team. However it was revealed in issue #93 the founding members were actually Skrulls in disguise.
• The Avengers #109 - Hawkeye is upset that The Scarlet Witch chooses The Vision over him and quits the team hoping to make it on his own.
• The Avenger #111-#112 - Daredevil and Black Widow are offered membership after helping The Avengers defeat Magneto. Daredevil declines but Black Widow accepts, only to leave an issue later!
• The Avengers #114 - The Swordsman (Jacques DuQuesne) rejoins The Avengers after changing his villainous ways. The main reason behind his change is due to him falling in love with Mantis who convinced him to give up his life of crime. At first The Avengers - especially Captain America do not trust the former villain and put him on probation but he soon proves a trustworthy ally. Mantis meanwhile, although not officially made an Avenger, starts going on missions with the team.
• The Avengers #137 - The Avengers find their numbers depleted after the death of The Swordsman, Captain America and Black Panther's self imposed absences and The Vision and Scarlet Witch being on honeymoon. Thus Moondragon is invited to join and summons are sent to all former members. The Wasp and Hank Pym (as Yellow Jacket) respond. The team also hold open try outs and former X-Man Beast is recruited.

The Avengers Annuals (1967)

• Avengers Annual #1 (Sep. 1967) - Original members Iron Man and Thor return as reserve members.

Uncanny Avengers (2012)

• Uncanny Avengers #1 (Oct. 2012) - As a result of AvX, Captain America and Thor started recruiting members for the Avengers Unity Squad a.k.a. The Uncanny Avengers to show the world that humans and mutants can work together. Havok was asked to lead the team and Wolverine was a natural selection being both an Avenger and on of the X-Men.
• Uncanny Avengers #4 (Feb. 2013) - After helping The Uncanny Avengers defeat Red Skull and his S-Men, Scarlet Witch and Rogue also join the team. Both characters played a key role in the first story arc and appeared from issue #1. However they weren't officially part of the team until issue #4.
• Uncanny Avengers #5 (Mar. 2013) - Wonder Man, Wasp and Sunfire join the team.

The New Avengers (2005)

• New Avengers #1-6 (Jan. 2005-June 2005) - The New Avengers form after the last team was disbanded 6 months earlier. The team started with Iron Man, Captain America, Spider-Man, Luke Cage, and Spider-Woman. Later added were Wolverine and Echo.

Avengers (2012)

• Avengers #1 (Dec. 2012) - This series was launched as part of the Marvel NOW! relaunch and the team started with the cast from the movie Marvel's The Avengers - Captain America, Iron Man, Thor, The Hulk, Hawkeye and Black Widow. In issue #1, Captain America and Iron Man decide The Avengers "must get bigger" to deal with the bigger threats in the horizon. The last page shows Captain America with a large number of new recruits after the rest of the team were captured. The recruitment drive is shown in the next issue.
• Avengers #2 (Dec. 2012) - Captain America and Iron Man had previously recruited a number of new members who were waiting in reserve. Now in order to rescue his captured team mates, Captain America calls the following heroes to join the core team: Cannonball, Captain Universe, Falcon, Hyperion, Manifold, Shang-Chi, Smasher, Sunspot and former members Captain Marvel, Spider-Man, Spider-Woman and Wolverine.

New Avengers (2013)

This series follows The Illuminati as opposed to any previous incarnation of New Avengers.

• New Avengers #1 (Jan. 2013) - After witnessing an event in Wakanda that can potentially lead to the end of the Universe, Black Panther calls together Captain America, Namor, Black Bolt, Mr. Fantastic, Iron Man and Doctor Strange to reform The Illuminati.
• New Avengers #24 (September 2014) - After an eight month timeskip, it is revealed that Bruce Banner, Amadeus Cho, Captain Britain and Yellowjacket have all joined the Illuminati.

Uncanny Avengers (2015)

• Uncanny Avengers #1 (October 2015) - After the events of AXIS, Captain America, Sunfire, Wasp, Havok, Wolverine, and Thor are no longer part of the team. They are replaced with Captain America (Sam Wilson), Brother Voodoo, Vision, Quicksilver and Sabretooth.

New Avengers (2015)

• New Avengers #1 (November 2015) - Eight months after Secret Wars, Sunspot unveils a new team of New Avengers consisting of himself, Songbird, Squirrel Girl, Hawkeye, Wiccan, Power Man, White Tiger, Hulkling and Pod.

Uncanny Avengers (2015)

• Uncanny Avengers #1 (November 2015) - Eight months after Secret Wars, Scarlet Witch, Captain America, Sabretooth and Vision have all left the team. The aged Steve Rogers rejoins the group, and recruits Deadpool, Synapse, Spider-Man, and the Human Torch to join Rogue, Quicksilver and Brother Voodoo.

All New All Different Avengers (2015)

• All-New All-Different Avengers #1 (November 2015) - Eight months after Secret Wars, the Avengers are no more and Avengers Tower is being sold off. After the Chitauri villain Warbringer attacks the Tower, Spider-Man (Miles Morales), Captain America (Sam Wilson) and Iron Man arrive to fight him.
• All-New All-Different Avengers #2 (December 2015) - Vision, Ms Marvel (Kamala Khan), Nova (Sam Alexander), and Thor (Jane Foster) arrive to assist the other heroes in battle.
• All-New All-Different Avengers #3 (January 2016) - With Warbringer defeated, the heroes choose to stay together and form the newest iteration of the Avengers.

In Other Media

Film

The Avengers: Age of Ultron

True to the comics, the second Avengers movie features a radical alteration of the team's line-up. The first movie features a team consisting of Iron Man, Captain America, Thor, the Hulk, Black Widow and Hawkeye, and this group is initially featured in the sequel. However, as the movie progresses, circumstances lead to the Hulk being declared MIA, Thor leaving to return to Asgard, Iron Man taking a leave of absence to spend more time with Pepper Potts, and Hawkeye returning home to help take care of his newborn son with his wife, Laura. The final shot of the movie reveals a new roster of Avengers consisting of Captain America, Black Widow, War Machine, Vision, Falcon and Scarlet Witch, with the group's base of operations now changed from Avengers Tower to the New Avengers Compound in upstate New York.