Many nonlinear evolution PDEs exhibit a finite time breakdown. Examples are the inviscid Burgers equation and the nonlinear heat equation, which can develop a shock and a blow-up singularity, respectively. One explanation of such phenomena involves the dynamics of singularities in the complex plane. When a singularity (pole or branch point) is located far into the complex plane, the solution appears smooth on the real axis. As the singularity travels towards the real axis, however, the solution profile steepens. Breakdown occurs when the singularity reaches the real axis. In this talk we shall review a few known results in this area, and discuss a numerical method based on Pade' approximation for computing the pole dynamics.