Vortices have recently received much attention in the research of planet formation, as they are believed to play a role in the formation of km-sized planetesimals by collecting dust particles in their centres. However, vortex dynamics is commonly studied in non-self-gravitating discs. The main goal here is to examine the effects of disc self-gravity on vortex dynamics. For this purpose, we employ the 2D shearing sheet approximation and numerically solve the basic hydrodynamic equations together with Poisson's equation to take care of disc self-gravity. A simple cooling law with a constant cooling time is adopted, such that the disc settles down into a quasi-steady gravitoturbulent state. In this state, vortices appear as transient structures undergoing recurring phases of formation, growth to sizes comparable to a local Jeans scale and eventual shearing and destruction due to the combined effects of self-gravity (gravitational instability) and background Keplerian shear.Each phase typically lasts about two orbital periods or less. As a result, in self-gravitating discs, the overall dynamical picture of vortex evolution is irregular consisting of many transient vortices at different evolutionary stages and, therefore, with various sizes up to the local Jeans scale. By contrast, in the non-self-gravitating case,long-lived vortex structures persist for hundreds of orbits via merging of smaller vortices into larger ones until eventually their size reaches the disc scale height. Vortices generate density waves during evolution,which turn into shocks. This phenomenon of wave generation by vortices is an inevitable consequence of the differential character (shear) of disc rotation. Therefore, the dynamics of density waves and vortices are coupled implying that, in general, one should consider both vortex and spiral density wave modes in order to get a proper understanding of self-gravitating disc dynamics.Our results suggest that given such an irregular and rapidly varying character of vortex evolution in self-gravitating discs, it may be difficult for such vortices to effectively trap dust particles in their centres, which is an alternative mechanism recently proposed for planetesimal formation. Further study of the behaviour of dust particles embedded in a self-gravitating gaseous disc is, however, required to strengthen this conclusion.