The divergence of new genes and proteins occurs through mutations that modulate protein function. However, mutations are pleiotropic and can have different effects on organismal fitness depending on the environment, as well as opposite effects on protein function and dosage. We review the pleiotropic effects of mutations. We discuss how they affect the evolution of gene and protein function, and how these complex mutational effects dictate the likelihood and mechanism of gene duplication and divergence. We propose several factors that can affect the divergence of new protein functions, including mutational trade-offs and hidden, or apparently neutral, variation.