The destabilization of a normal mode of the uniform stationary state of a spatially extended system may initiate a whole cascade of bifurcations of families of states of increasing complexity. We discuss methods for the linear-stability analysis of the new solutions based on symmetry and continuity requirements which may be applied when no explicit analytic representation is available. We always find a family of simply periodic traveling-wave states bifurcating from the uniform stationary state (primary bifurcation). Stabilization of these simply periodic states is connected with a secondary bifurcation of a family of doubly periodic states, which upon stabilization may in turn be connected with a tertiary bifurcation of triply periodic states, etc. Each of the families may contain solitary states as limiting cases. The general theory is applied to a few representative examples. © 1981 The American Physical Society.