When analyzing large networks of coupled oscillators, it is often beneficial to consider the limit as the number of oscillators goes to infinity. Continuum limit is essential for understanding such phenomena as the onset of synchronization in the Kuramoto model of coupled phase oscillators with random intrinsic frequencies, emergence and bifurcations of chimera states, and stability of some other spatiotemporal patterns. In this talk, we discuss derivation and rigorous justification of the continuum limit for the Kuramoto model on convergent families of deterministic and random graphs. The latter include Erdos-Renyi, small-world, and power law graphs.