Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. Recent advances have allowed for a better comprehension of synchronization phenomena even when oscillating elements are constrained to interact in a complex topology. In our group, we have been studying synchronization processes on top of complex networks, unraveling new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem have been carried out. Finally, its is also our goal to apply our findings to several different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.