Completing Achievements gives you rewards in the form of Experience and Gems. Your Achievements will also appear in the Game Center and on your Profile; each Achievement gives you a set amount of Game Center points.
Since the "Player Profile" update, player's achievements/status (including overachievements) are made public. The 19/12/16 update replaced the Achievements button with the Events button, making Achievements only visible through Player Profile. However, some events function like mini-achievements with limited time period.
There are 31 Achievements currently available. The "Keep your village safe" has only one level with gives all three stars, the others each has three levels, for a total of 91 levels:
Home Village Achievents
Bigger Coffers
Upgrade a Gold Storage to level 2: 10 
/ 2 
Upgrade a Gold Storage to level 5: 100 / 5
Upgrade a Gold Storage to level 10: 1,000 / 10
Get those Goblins!
Win 10 stars on the Campaign Map: 10 / 5
Win 50 stars on the Campaign Map: 100 / 10
Win 150 stars on the Campaign Map: 1,000 / 20
Bigger & Better
Upgrade Town Hall to level 3: 10 / 5
Upgrade Town Hall to level 5: 100 / 10
Upgrade Town Hall to level 8: 1,000 / 20
Nice and Tidy
Remove 5 Obstacles (trees, rocks, bushes): 10 / 5
Remove 50 Obstacles (trees, rocks, bushes): 100 / 10
Remove 500 Obstacles (trees, rocks, bushes): 1,000 / 20
Release the Beasts
Unlock Archer in the Barracks: 10 / 5
Unlock Wall Breaker in the Barracks: 100 / 10
Unlock Dragon in the Barracks: 1,000 / 20
Gold Grab
Steal 20,000 Gold: 10 / 5
Steal 1,000,000 Gold: 100 / 10
Steal 100,000,000 Gold: 1,000 / 20
Elixir Escapade
Steal 20,000 Elixir: 10 / 5
Steal 1,000,000 Elixir: 100 / 10
Steal 100,000,000 Elixir: 1,000 / 20
Sweet Victory!
Achieve a total of 75 Trophies in Multiplayer battles: 10 / 5
Achieve a total of 750 Trophies in Multiplayer battles: 100 / 10
Achieve a total of 1,250 Trophies in Multiplayer battles: 1,000 / 450
Empire Builder
Rebuild the Clan Castle: 10 / 5
Upgrade Clan Castle to Level 2: 100 / 10
Upgrade Clan Castle to Level 4: 1,000 / 20
Wall Buster
Destroy 10 Walls in Multiplayer battles: 10 / 5
Destroy 100 Walls in Multiplayer battles: 100 / 10
Destroy 2,000 Walls in Multiplayer battles: 1,000 / 20
Humiliator
Destroy 10 Town Halls in Multiplayer battles: 10 / 5
Destroy 100 Town Halls in Multiplayer battles: 100 / 10
Destroy 2,000 Town Halls in Multiplayer battles: 1,000 / 50
Union Buster
Destroy 25 Builder's Huts in Multiplayer battles: 10 / 5
Destroy 250 Builder's Huts in Multiplayer battles: 100 / 10
Destroy 2,500 Builder's Huts in Multiplayer battles: 1,000 / 30
Conqueror
Win 25 Multiplayer battles: 10 / 5
Win 250 Multiplayer battles: 100 / 10
Win 5,000 Multiplayer battles: 1,000 / 20
Unbreakable
Successfully defend against 10 attacks: 10 / 5
Successfully defend against 250 attacks: 100 / 50
Successfully defend against 5,000 attacks: 1,000 / 100
Friend in Need
Donate 100 Clan Castle capacity worth of reinforcements: 10 / 5
Donate 5,000 Clan Castle capacity worth of reinforcements: 100 / 25
Donate 25,000 Clan Castle capacity worth of reinforcements: 1,000 / 250
Mortar Mauler
Destroy 25 Mortars in Multiplayer battles: 10 / 5
Destroy 500 Mortars in Multiplayer battles: 100 / 10
Destroy 5,000 Mortars in Multiplayer battles: 1,000 / 20
Heroic Heist
Steal 20,000 Dark Elixir: 10 / 5
Steal 250,000 Dark Elixir: 100 / 10
Steal 1,000,000 Dark Elixir: 1,000 / 20
League All-Star
Join the Crystal League: 100 / 250
Reach the Master League: 500 / 1,000
Become a Champion!: 2,000 / 2,000
X-Bow Exterminator
Destroy 1 X-Bow in a Multiplayer battle: 50 / 50
Destroy 250 X-Bows in Multiplayer battles: 100 / 100
Destroy 2,500 X-Bows in Multiplayer battles: 1,000 / 200
Firefighter
Destroy 10 Inferno Towers in Multiplayer battles: 50 / 100
Destroy 250 Inferno Towers in Multiplayer battles: 500 / 200
Destroy 5,000 Inferno Towers in Multiplayer battles: 5,000 / 1,000
War Hero
Score 10 stars for your clan in War Battles: 50 / 50
Score 150 stars for your clan in War Battles: 500 / 200
Score 1,000 stars for your clan in War Battles: 5,000 / 1,000
Treasurer
Collect 800,000 Gold from the Treasury: 100 / 25
Collect 15,000,000 Gold from the Treasury: 500 / 100
Collect 100,000,000 Gold from the Treasury: 5,000 / 500
Anti-Artillery
Destroy 20 Eagle Artillery in Multiplayer battles: 100 / 150
Destroy 200 Eagle Artillery in Multiplayer battles: 800 / 300
Destroy 2000 Eagle Artillery in Multiplayer battles: 5,000 / 1,000
Sharing is Caring
Donate 100 spell storage worth of Spells: 20 / 20
Donate 2000 spell storage worth of Spells: 200 / 100
Donate 10000 spell storage worth of Spells: 2,000 / 500
Keep your village safe
Connect to Game Center, Google Play, QQ or WeChat: 100 / 50
Builder Base Achievements
Master Engineering
Upgrade Builder Hall to level 3: 20 / 10
Upgrade Builder Hall to level 5: 200 / 30
Upgrade Builder Hall to level 8: ? / ?
Next Generation Model
Unlock Sneaky Archer in the Builder Barracks: 20 / 10
Unlock Cannon Cart in the Builder Barracks: 200 / 30
Unlock Super P.E.K.K.A in the Builder Barracks: ? / ?
Un-Build It
Destroy 5 Builder Halls in Versus Battles: 20 / 10
Destroy 100 Builder Halls in Versus Battles: 200 / 30
Destroy 2,000 Builder Halls in Versus Battles: ? / ?
Champion Builder
Donate 100 spell storage worth of Spells: 20 / 20
Donate 2000 spell storage worth of Spells: 200 / 100
Donate 10000 spell storage worth of Spells: 2,000 / 500
Sharing is Caring
Donate 100 spell storage worth of Spells: 20 / 20
Donate 2000 spell storage worth of Spells: 200 / 100
Donate 10000 spell storage worth of Spells: 2,000 / 500
Sharing is Caring
Donate 100 spell storage worth of Spells: 20 / 20
Donate 2000 spell storage worth of Spells: 200 / 100
Donate 10000 spell storage worth of Spells: 2,000 / 500
Total
Completing all Home Village Achievements gives a total of 10,807 Gems and 43,540 Experience, which is enough to buy all five Builder's Huts (with over 7,000 gems left over).
Completing the entire League All-Star achievement totals 3,250 Gems, which is only 250 gems away from the price of the third, fourth, and fifth Builder's Huts.
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