The atmosphere of the Sun is being stirred and reshaped continuously through the emergence of magnetized plasma from the convection zone. Magnetic elements coming from the solar interior produce structures in the photosphere, chromosphere and corona in a bewildering range of length and time scales, from the smallest magnetic tubes to the largest active regions. Although still at an early stage, the computer simulation of these intrinsically three-dimensional processes can already provide important physical insight and explain some of the observational results obtained with ground-based detectors and in space. This review summarizes the challenge posed by flux emergence processes to numerical simulation, discusses why the magnetic plasma can rise through the photosphere in spite of the latter's subadiabatic stratification and presents a number of results on active region formation at the photosphere obtained in recent years.