  We decompose sphere partition functions and indices of three-dimensional N=2
gauge theories into a sum of products involving a universal set of "holomorphic
blocks". The blocks count BPS states and are in one-to-one correspondence with
the theory's massive vacua. We also propose a new, effective technique for
calculating the holomorphic blocks, inspired by a reduction to supersymmetric
quantum mechanics. The blocks turn out to possess a wealth of surprising
properties, such as a Stokes phenomenon that integrates nicely with actions of
three-dimensional mirror symmetry. The blocks also have interesting dual
interpretations. For theories arising from the compactification of the
six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a
basis of wavefunctions in analytically continued Chern-Simons theory on M. For
theories engineered on branes in Calabi-Yau geometries, the blocks offer a
non-perturbative perspective on open topological string partition functions.
