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