  We consider dynamical systems depending on one or more real parameters, and
assuming that, for some ``critical'' value of the parameters, the eigenvalues
of the linear part are resonant, we discuss the existence -- under suitable
hypotheses -- of a general class of bifurcating solutions in correspondence to
this resonance. These bifurcating solutions include, as particular cases, the
usual stationary and Hopf bifurcations. The main idea is to transform the given
dynamical system into normal form (in the sense of Poincar\'e-Dulac), and to
impose that the normalizing transformation is convergent, using the convergence
conditions in the form given by A. Bruno. Some specially interesting
situations, including the cases of multiple-periodic solutions, and of
degenerate eigenvalues in the presence of symmetry, are also discussed with
some detail.
