In this paper we define and study digital manifolds of arbitrary dimension, and provide (in particular)a general theoretical basis for curve or surface tracing in picture analysis. The studies involve properties such as one-dimensionality of digital curves and (n-1)-dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in continuous topology. The presented approach is fully based on the concept of adjacency relation and complements the concept of dimension as common in combinatorial topology. This work appears to be the first one on digital manifolds based ona graph-theoretical definition of dimension. In particular, in the n-dimensional digital space, a digital curve is a one-dimensional object and a digital hypersurface is an (n-1)-dimensional object, as it is in the case of curves and hypersurfaces in the Euclidean space. Relying on the obtained properties of digital hypersurfaces, we propose a uniform approach for studying good pairs defined by separations and obtain a classification of good pairs in arbitrary dimension. We also discuss possible applications of the presented definitions and results.