We investigate the stability and dynamics of nonlinear structures that arise from the Kadomtsev–Petviashvili equation. For initially flat propagating cnoidal waves and solitons, the region of instability in perturbed wavevector K–space is found. This is done by combining information obtained both by approximate and exact calculations. Growth rates are found. Interesting analogies with the deep–water wave problem are pointed out. It might seem strange that this analysis should be necessary when the problem has been ‘solved’ exactly by Kuznetsov and co–workers. However, as those authors admit, their treatment is extremely difficult to apply to any but a few specific limits. So much so that they miss a stability boundary even in one of those limits. We will comment on this. Here, by combining all our results, we are able to present a reasonably complete picture. Formulae are explicit and simple. Once the instability develops, two–dimensional ‘lumps’ are produced, as well as a new diminished cnoidal wave. We will concentrate on the lumps and their further fate. The two–dimensional solitons or lumps are then investigated in three dimensions by numerical simulations. A distortion in a plane perpendicular to the motion leads to an oscillation of the lumps. This result is in contradistinction to the result of a perturbation in a plane containing the direction of motion. In that simulation of Senatorski and Infeld, lumps were destroyed completely and three–dimensional solitons were formed. Thus a link was found between two entities that were hitherto considered separately. In all, we have an initially two–dimensional soliton that can either produce a three–dimensional decay product or else exhibit a new oscillation. This will depend on the plane of the perturbation.