Synchronization phenomena are abundant in nature, science, engineering and social life, such as in organ pipes, fireflies and even in the mechanics of bridges. But synchronization was first recognized by Christiaan Huygens in 1665 for coupled pendulum clocks; this was the beginning of nonlinear sciences. In the last two decades, this concept has been successfully extended to more complex systems, as coupled chaotic oscillators and even distributed systems as complex networks. Complex networks were first studied by Leonhard Euler in 1736 when he solved the Königsberger Brückenproblem. Recent research has revealed a rich and complicated network topology in various model systems as well as in several fields of applications, such as transportation and social networks, or the WWW. We will discuss synchronization phenomena on active complex networks. We will report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We will also present new emergent features coming out from the interplay between the structure and the function of the underlying patterns of connections. The classic local stability in dynamic networks is compared to the new basin stability concept. Finally, several applications of synchronization in complex networks from human brain via power grids to climate are given.