The continuum limit is an approximate procedure, by which coupled dynamical systems on large graphs are replaced by an evolution integral equation on a continuous spatial domain. This approach has been instrumental for studying dynamics of diverse networks throughout physics and biology. We use the ideas from the theories of graph limits and nonlinear evolution equations to develop a rigorous justification for using the continuum limit in a variety of dynamical models on deterministic and random graphs. As a specific application, we discuss synchronization in small-world networks of Kuramoto oscillators.