The paper deals with coding problems arising in symbolic dynamics for certain shift-invariant spaces (called shift spaces) of infinite sequences over a finite alphabet. Shift spaces are defined by constraints on the finite words that are allowed to appear in the sequences. The emphasis is on shift spaces (called sofic shifts) defined by finite directed labelled graphs. Much attention is given to spaces of sequences with group structure (group shifts) and, more generally, to spaces of sequences which are homogeneous in the sense that they are acted upon transitively by a group shift (homogeneous shifts). Various results on sofic, group, and homogeneous shifts are obtained. The central problem treated in the paper is the classification of homogeneous shifts up to conjugacy. It is shown that (1) every homogeneous shift is conjugate to the product of a full shift (a completely unconstrained set of sequences over a finite alphabet) and a finite shift (a shift-invariant set consisting of only finitely many periodic sequences); (2) every mixing homogeneous shift is conjugate to a full shift. Some results on the future cover of a homogeneous shift are presented. Improved procedures to test whether a given shift space is a homogeneous shift are suggested.