Going back to Henri Poincaré, the main concern of the theory of dynamical systems for differential equations is the qualitative characterization of solutions. Symmetries, described by group transformations, help immensely in this quest --- providing that they exist, which is often the case only in very special dynamical systems. In this thesis, we significantly enlarge the class of dynamical systems which can be studied by symmetry methods, moving our focus from groups to groupoids as the underlying algebraic structure describing symmetry. Building on the groupoid framework, we fundamentally generalize the notion of equivariance and equivariant bifurcation theory. In summary, we present a new unified theory of symmetric spatio-temporal patterns.
