Here is the full list of all 36 Besiege achievements.
Complete Tree of Akhmora without using Crossbows, Cannons and Explosives
Complete Kahraz Village without using Cannons, Crossbows, Drills or Saws
Lift the boulder in Stock Tower 10 units off the ground
Complete Towering Eye without ever being hit
Complete Krolmar
Beat Old Howl Battlefield while having blown up all the bombs
Complete the Revolving Monolith without using Grabbers
Complete Old Mining Site without pressing a key
Complete the Campaign
Complete the Mountain Barrier without explosives
Complete Perimeter Wall without pressing a key
Beat the Consumed King without using mechanical blocks
At least 2 players joined your server
Conquer Valfross
Join over 10 servers
Beat the Southern Shrine by primarily using fire to kill the birds
Complete a player-made map
Reach enough altitude to freeze your machine
Ring all 3 Awakening Bells at the same time
Beat Queens Fodder without using fire or explosives
Conquer Tolbrynd
Destroy the Duke's freighters without flying blocks
Steal the Duke's knowledge in under 2 minutes!
Build your first level using the level editor
Share your machine with the whole world
Beat Scouts of Tolbrynd without cannons, flamethrowers or crossbows
Complete a campaign level with under 3 blocks
Put out a burning block using water
Get hit by lightning
Conquer Ipsilon
Defeat 1000 AI units
Subscribe and download someone else's machine
Complete a campaign level under 2 seconds
Complete Standing Stone without using armor
Travel at speeds faster than is sensible
Set yourself on fire