The authors develop a theory of bifurcations of differential equations with Lie point symmetries. Viewing the differential equation as an ``algebraic equation'' on some jet bundle is the key of the analysis. The authors show how the well-known results of equivariant bifurcation theory can be formulated in this context and how the results carry over. Especially using a Lyapunov-Schmidt reduction for steady state bifurcation problems it is shown that the symmetry algebra of the reduced equation is a subalgebra of the symmetry algebra of the full equation. Moreover, the precise relation between these two algebras is given. For ODE's relations between the existence of bifurcation points and the Lie point symmetries are discussed. The author indicates how the results apply to the case of Hopf bifurcations. Applying these results to PDE's leads to some technical complications. The authors discuss these problems and presents some ideas how to solve them.