Spacetimes representing point particles and black holes in 2+1-dimensional gravity with a negative cosmological constant can all be represented as cut-and-paste operations on 2+1-dimensional anti-de Sitter space. This means that processes like making a spinning black hole from two point particles colliding with an impact parameter can be described in closed form. I will review the geometry of this, much of it going back to the mid-90s, and talk about my own ongoing work.