The entropy of discrete random variables can be harder to bound and control than their continuous counterparts. In particular, a number of functional inequalities and arguments based on scaling do not pass over, meaning that results such as a discrete entropy power inequality are elusive. I will review results from a number of papers, including a maximum entropy result for Poisson random variables, a form of discrete entropy concavity and a Poincare inequality for integer-valued random variables. Many of these results rely on the interplay between ultra-log-concavity (which gives a discrete Bakry-Emery criterion) and Renyi's thinning operation.