The continuity conditions and the transmission conditions involved in domain decomposition methods are of major importance for the fast and robust convergence of these algorithms. In this review paper, we study step by step the methodology to derive various continuity conditions and compatible transmission conditions for two different types of problems, i.e. the Laplace equation and the Helmholtz equation. An original homogeneous formulation is also presented both in the continuous and in the discrete analysis. Numerical experiments show the relative efficiency of these continuity conditions and of these transmission conditions on academic problems.