The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat-Tits building of SL_d(K) and techniques for computing with apartments and membranes. The original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry. This talk will include an introduction to tropical convexity and Bruhat-Tits buildings. This is joint work with Michael Joswig and Bernd Sturmfels.