Mrk 421 is a high-synchrotron-peaked blazar featuring bright and persistent GeV and TeV emission. We use multi-wavelength light curves of Mrk 421 spanning 5.5 years with FACT (TeV) and Fermi LAT (GeV) in the gamma rays, Swift BAT, Swift XRT and MAXI in the X-rays, together with optical and radio data and investigate the physical processes driving the emission and variability. Observations by FACT are continuous and not triggered, so the source was found in a wide range of flux states and more than 30 flares were identified. The TeV and X-ray light curves feature very similar flares with rise and decay times of a few days and zero lag, characteristic of electron processes. At least two parameters per flare, the amplitude and the cut-off energy, are required to explain the observed variability. The GeV light curve leads and is strongly correlated with the optical and radio light curves as expected from SSC emitting shock propagating in a conical jet.