A common theme among previously proposed models for network epidemics is the assumption that the propagating object (e.g., a pathogen [in the context of infectious disease propagation] or a piece of information [in the context of information propagation]) is transferred across network nodes without going through any modification or evolutionary adaptations. However, in real-life spreading processes, pathogens often evolve in response to changing environments and medical interventions, and information is often modified by individuals before being forwarded. In this article, we investigate the effects of evolutionary adaptations on spreading processes in complex networks with the aim of 1) revealing the role of evolutionary adaptations on the threshold, probability, and final size of epidemics and 2) exploring the interplay between the structural properties of the network and the evolutionary adaptations of the spreading process.