Computational seismology is entering a new era. The reason is, that with the current supercomputer technology, the frequency band in which seismic waves are observed following regional or global earthquakes, can be simulated numerically for realistic 3D earth models for the first time. Depending on the spatial scales under consideration (whole planet, a sedimentary basin at risk from local earthquakes, a volcano with high risk for future eruptions) this will lead to considerable improvement (1) in the understanding of the structural properties (e.g. the Earth's mantle, the inside of a sedimentary basin or a volcano) and (2) in forecasting strong ground motion for realistic earthquake scenarios. The latter point may have considerable long-term societal benefits, as we now understand that the short-term prediciton of large earthquakes may never be possible. We will discuss the current hot topics in computational seismology and show applications to earthquake scenario simulations, dynamic rupture problems, global wave propagation and volcano seismology.