Structure-preserving discretization in the field of geometry is the paradigm of discrete differential geometry. In some aspects, the discrete theory turns out to be even richer than its smooth counterpart. It focuses on developing constructive methods. The well-established theory of discrete conformal maps and circle patterns is related to discrete integrable models of mathematical physics and has found applications in geometry processing. We present their generalizations beyond the conformal limit: decorated discrete conformal maps and ring patterns, which share the corresponding existence and uniqueness statements. The theory and construction methods are based on convex variational principles. We also briefly explain how structure preserving discretizations recently helped to answer the long-standing question whether a compact surface in three-space is uniquely determined by its metric and curvatures (Bonnet problem).