In this work, the picosecond laser ablation process of aluminium nitride (AlN) and silicon nitride (Si3N4) ceramics is studied. The laser irradiation is performed by a Nd:YAG laser system that generates pulses with a duration of 10 ps and operates at the wavelengths of 266 nm, 355 nm, 532 nm, and 1064 nm. The ablation phenomena characterization is based on analyses of the ablation depth, the surface morphology and the chemical changes induced by the laser radiation. The dependences are studied of these characteristics on the processing parameters, namely, the laser wavelength, fluence and number of applied pulses, and the ambient pressure. The results presented concern the drilling geometry and the tracks formed when the sample is moving with respect to the laser beam. The ablation efficiencies in terms of ablation depth per incident fluence and per pulse are also discussed. It is found that the laser treatment results in a variety of micro- and nanostructures on the surface of the material. Ripple structure formation is observed in both ceramics, which are of different type with respect to the period and orientation. The ablation mechanisms are considered and evidence is provided that in picosecond regime decomposition of the ceramics could takes place with aluminium or silicon formation on the surface. The paper offers novel data on the fundamental picture of picosecond laser-matter interaction that can be useful in ceramic surface structuring and in designing methods for fabricating complex composite materials.