This chapter provides a comprehensive overview of the pivotal roles of adaptation and synchronization in complex systems, emphasizing their mathematical underpinnings while also identifying areas where a complete theoretical framework remains to be established. Defining a complex system precisely remains challenging, yet one useful approach characterizes complex systems by their capacity to produce emergent collective behavior through the interactions of numerous interdependent components. Importantly, such collective behavior cannot be attributed solely to the nonlinearity of individual components or the heterogeneity of their interactions. Instead, it emerges from a nuanced interplay between the properties of individual elements and the network architecture, resulting in phenomena such as spontaneous order and self-organization.