Model complexity of Genetic Programming (GP) as a learning machine is currently attracting considerable interest from the research community. Here we provide an up-to-date overview of the research concerning complexity measure techniques in GP learning. The scope of this review includes methods based on information theory techniques, such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC); plus those based on statistical machine learning theory on generalization error bound, namely, Vapnik-Chervonenkis (VC) theory; and some based on structural complexity. The research contributions from each of these are systematically summarized and compared, allowing us to clearly define existing research challenges, and to highlight promising new research directions. The findings of this review provides valuable insights into the current GP literature and is a good source for anyone who is interested in the research on model complexity and applying statistical learning theory to GP.