Since the discovery of dissipative Kerr solitons in optical microresonators, significant progress has been made in the understanding of the underlying physical principles from the fundamental side and generation of broadband coherent optical Kerr frequency combs from the applied side. Rich nonlinear dynamics of the discovered coherent dissipative structures have been explored and widely applied from distance measurements and telecommunication to neuromorphic optical computing. However, these studies were mostly limited to the single-resonator case, in which the nonlinear dynamics is essentially one-dimensional. On the other hand, increasing the number of particles (i.e., resonators) and creating new dimensions in photonic devices are expected to provide a plethora of novel dynamical effects with a fundamental and technological potential, which however remains an uncharted territory, with the large capacity for both theoretical and experimental explorations. With this thesis, we explore this direction by investigating the nonlinear dynamics in various lattices of nonlinear optical microresonators, extending the conventional single-resonator paradigm. We consider two types of photonic lattices: synthetic and spatial. Providing the analytical, numerical, and experimental studies, we investigate emerging four-wave mixing processes, chaotic states, and formation of coherent structures. In the synthetic frequency dimension framework, we consider electro-optically and dispersion modulated resonators, demonstrating the formation of novel nonlinear states as well and related new four-wave mixing pathways that result in the spectral broadening of frequency combs. In the case of coupled resonators, we investigate parametric processes, existence and stability of coherent structures and demonstrate potential applications for optical parametric oscillators and microwave signal generation. We also develop a general theory of nonlinear dynamics and Kerr frequency comb formation in lattices of resonators, demonstrating the multidimensional nature of the nonlinear processes in such systems. We investigate in detail the two-dimensional spatio-temporal dynamics in chains of equally coupled resonators. Finally, we describe an open-source software -- PyCORe, developed during the course of this thesis, which allows simulation of nonlinear dynamics in the systems under consideration.