Controlled direct acceleration of electrons in very strong laser fields can offer a path towards ultra-compact accelerators. Such a direct acceleration requires rectification and decoupling of the oscillating electromagnetic laser field from the electrons in a suitable way. Researchers worldwide try to tackle this challenge. In experiments at the Max Born Institute, direct laser acceleration of electrons could now be demonstrated and understood in detail theoretically. This concept is an important step towards the creation of relativistic and ultra-short electron pulses within very short acceleration distances below one millimeter. Resulting compact electron and related x-ray sources have a broad spectrum of applications in spectroscopy, structural analysis, biomedical sciences and for nanotechnology.