D-branes at singularities give rise to quiver gauge theories on their world-volume, which may be studied in the context of AdS/CFT. For toric spaces the four-dimensional D-brane gauge theories reduce to two-dimensional lattice models on a bipartite torus graph, called brane tilings. I will review the derivation of the brane tilings and their origin in string theory, and discuss two statistical mechanical applications: spin chain models of operator mixing, and a fermionic representation of the random dimer model for the topological string partition function. \\ \par \noindent Note: This seminar will have a pedagogical and a technical part. Also part of the joint IHES-CEA/Saclay seminar series ``Black Holes, Topological Strings, Pure Spinors, Gauge Theorie''