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Mean curvature equations of general quasilinear type in connection with contact-angle boundary conditions are considered in this paper. We investigate the existence, uniqueness and continuous dependence of the solution in classical function spaces. On the one hand, a survey of techniques and ideas developed in the 1970s and 1980s, mainly by Uraltseva, is presented. On the other hand, extensions of these results are also proposed: we formulate growth conditions for the general dependence of the potential on the -variable, and we extend the existence and uniqueness statements to this case. Moreover, the regularity assumptions on the right-hand side are relaxed, and alternative proofs for the higher-order estimates and the existence result are provided.