Modeling and understanding complex non-manifold shapes is a key issue in shape analysis and retrieval. The topological structure of a non-manifold shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we consider a representation for arbitrary shapes, that we call Manifold-Connected Decomposition (MC-decomposition), which is based on a unique decomposition of the shape into nearly manifold parts. We present efficient and powerful two-level representations for non-manifold shapes based on the MC-decomposition and on an efficient and compact data structure for encoding the underlying components. We describe a dimension-independent algorithm to generate such decomposition. We also show that the MC-decomposition provides a suitable basis for geometric reasoning and for homology computation on non-manifold shapes. Finally, we present a comparison with existing representations for arbitrary shapes.