Kimura’s neutral theory of evolution has inspired researchers from the evolutionary computation community to incorporate neutrality into Evolutionary Algorithms (EAs) in the hope that it can aid evolution. The effects of neutrality on evolutionary search have been considered in a number of studies, the results of which, however, have been highly contradictory. In this paper, we analyse the reasons for this and we make an effort to shed some light on neutrality by addressing them. We consider two very simple forms of neutrality: constant neutrality — a neutral network of constant fitness, identically distributed in the whole search space — and bit-wise neutrality, where each phenotypic bit is obtained by transforming a group of genotypic bits via an encoding function. We study these forms of neutrality both theoretically and empirically (both for standard benchmark functions and a class of random MAX-SAT problems) to see how and why they influence the behaviour and performance of a mutation-based EA. In particular, we analyse how the fitness distance correlation of landscapes changes under the effect of different neutral encodings and how phenotypic mutation rates vary as a function of genotypic mutation rates. Both help explain why the behaviour of a mutation-based EA may change so radically as problem, form of neutrality and mutation rate are varied.