Your achievements and stats can be viewed in Steam under: "Community -> My profile -> View all my games -> Team Fortress 2 (view stats)"
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Dynasty → Win 20 games.

Flamethrower → Set five enemies on fire in 30 seconds.

Grey Matter → Get 25 headshots as a Sniper.

Hard to Kill → Get five kills in a row without dying.

Hardcore → Accumulate 1000 total kills.

Head of the Class → Play a complete round with every class.

Impenetrable Defense → Successfully defend Dustbowl without giving up a capture.

Impossible Defense → Successfully defend Gravel Pit without giving up a capture.

Lightning Offense → Win Well in 5 minutes or less.

Master of Disguise → Trick an opposing medic into healing you.

Nemesis → Get five revenge kills.

Powerhouse Offense → Win 2Fort with a shutout.

Relentless Offense → Win Hydro without giving up a capture.

Sentry Gunner → Accumulate 10 sentry gun kills with a single turret.

Team Doctor → Accumulate 25000 heal points as a Medic.

With Friends Like these... → Play in a game with 7 or more players from your friends list.

World Traveler → Play a complete game on 2Fort, Dustbowl, Granary, Gravel Pit, Hydro, and Well (CP).