Many adaptive evolutionary systems display spatial and temporal features, such as long-range correlations, typically associated with the critical point of a phase transition in statistical physics. Empirical and theoretical studies suggest that operating near criticality enhances the functionality of biological networks, such as brain and gene networks, in terms for instance of information processing, robustness, and evolvability. While previous studies have explained criticality with specific system features, we still lack a general theory of critical behavior in biological systems. Here we look at this problem from the complex systems perspective, since in principle all critical biological circuits have in common the fact that their internal organization can be described as a complex network. An important question is how self-similar structure influences self-similar dynamics. Modularity and heterogeneity, for instance, affect the location of critical points and can be used to tune the system toward criticality. We review and discuss recent studies on the criticality of neuronal and genetic networks, and discuss the implications of network theory when assessing the evolutionary features of criticality.