A spline space suitable for Isogeometric Analysis (IgA) on multi-patch domains is presented. Our construction is motivated by emerging requirements in isogeometric simulations. In particular, IgA spaces should allow for adaptive mesh refinement and they should guarantee the optimal smoothness of the discretized solution, even across interfaces of adjacent patches. Given a domain manifold M consisting of individual patches (isomorphic to the unit square or cube) that are glued together along interfaces, we present a construction of multi-patch B-splines defined on them. Their smoothness is enhanced by locally modifying or merging basis functions around the boundary of each patch. The resulting multi-patch B-splines with enhanced smoothness (MPBES) possess the property of local linear independence and form a nonnegative partition of unity. Moreover, their span can be characterized as the linear space of all piecewise polynomial functions on the domain manifold that possess certain smoothness properties. Subsequently, adaptively refined MPBES are obtained by generalizing the construction of truncated hierarchical (TH) B-splines. More precisely, a nested sequence of spaces spanned by MPBES is considered, corresponding to steps of local enrichment. In addition, an inversely nested sequence of subdomains (which are submanifolds of M) is used to specify the local refinement level of functions in these spaces. Finally, truncated hierarchical MPBES are obtained by means of the selection and truncation mechanism of THB-splines. The desired properties of linear independence and convex partition of unity are maintained. The paper presents several numerical examples which demonstrate potential applications of the new basis in isogeometric analysis.