The overall project objectives are to produce new knowledge in the area of codim k bifurcations for continuous and discrete (smooth and non-smooth) dynamical systems and provide training in this area of research to early stage researchers. More exactly, we plan firstly to study degenerate two-dimensional Bautin bifurcation for the case when the second Lyapunov coefficient equals zero. Secondly, we aim to study degenerate four-dimensional Hopf-Hopf bifurcations. The degeneracy arises in this case when one or more of the nine generic conditions needed in obtaining a normal form fail to be satisfied. The third and fourth objectives are to study other codim k bifurcations in smooth and non-smooth dynamical systems arising from other bifurcations which bear or not a known name in the literature. In particular, we will focus on discontinuous piecewise differential systems, respectively, continuous and discrete non-smooth dynamical systems resulting from modelling oscillators with impacts. A number of about 40 researchers (30 ERs and 10 ESRs) will contribute to achieving the project objectives by a networking approach based on about 225 months of secondments. Two types of secondments are planned, one for research and another for training. The training is of type training-through-research. During the research secondments, the project ERs will perform research for achieving the planned project tasks and will support ESRs on their training. The training secondments are dedicated to ESRs. Apart from the planned training, the ESRs will participate also to the project research. They will receive specific research tasks from their local managers to contribute to the project research objectives.