A magnetic vortex and an antivortex can annihilate, resulting in a homogeneous magnetization. A detailed description of the magnetization dynamics of such annihilation processes is obtained by micromagnetic simulations based on the Landau-Lifshitz-Gilbert equation. We show that, depending on the relative polarization of the vortex-antivortex pair, the annihilation process is either a continuous transformation of the magnetic structure or it involves the propagation of a micromagnetic singularity (Bloch point) causing a burstlike emission of spin waves. These results provide new insight into a fundamental micromagnetic process that has recently been proposed for a controlled generation of spin waves.