A fundamental result in 3-dimensional contact topology due to Ding-Geiges tells us that any contact 3-manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3-sphere. So it's natural to ask how simple or complicated a surgery diagram could be for a particular contact manifold? Contact surgery number is a measure of this complexity. In this talk, I will define this notion of complexity and discuss some examples. This is joint work with Marc Kegel.