Network theory has proven a powerful and general framework for studying the effects of complex interaction topology on the dynamics of many real systems in biology, physics and the social sciences, which show a variety of heterogeneous and multi-scale connectivity patterns. Although much work has been done in this field, many open questions remain about what role network topology plays in influencing the behaviors of complex systems. This dissertation examines the effects of complex network structure on the formation of collective oscillations and waves. In particular we study the propagation of epidemic fronts in multi-scale networks, the interplay between mutual and driven synchronization in heterogeneous oscillator networks, and the emergence of collective transport waves in driven randomly pinned oscillator networks. Qualitatively new behavior is found, and new reduction and analysis techniques are developed which allow us to understand the relationship between connectivity structure and the dynamics in these processes. Broadly, this work makes unique contributions to the exploration of fully non-equilibrium pattern formation and nonlinear dynamics in complex networks.