The description of atmospheric processes generally leads to systems of stiff differential equations. For such systems explicit integration methods are only stable for very small time steps. For this purpose implicit-explicit (IMEX) methods were developed. The most important components of these IMEX methods are suitable algorithms for large systems of linear equations that make use of the special sparse structures of the problems. A further gain in efficiency is obtained by so-called multi-rate procedures, which allow the use of different time step sizes in separate model regions and for different processes.