Session types are a well-established framework for the specification of interactions between components of a distributed systems. An important issue is how to determine the type for an open system, i.e., obtained by assembling subcomponents, some of which could be missing. To this end, we introduce partial sessions and partial (multiparty) session types. Partial sessions can be composed, and the type of the resulting system is derived from those of its components without knowing any suitable global type nor the types of missing parts. To deal with this incomplete information, partial session types represent the subjective views of the interactions from participants’ perspectives; when sessions are composed, different partial views can be merged if compatible, yielding a unified view of the session. Incompatible types, due to, e.g., miscommunications or deadlocks, are detected at the merging phase. In fact, in this theory the distinction between global and local types vanishes. We apply these types to a process calculus for which we prove subject reduction and progress, so that well-typed systems never violate the prescribed constraints. In particular, we introduce a generalization of the progress property, in order to accommodate the case when a partial session cannot progress not due to a deadlock, but because some participants are still missing. Therefore, partial session types support the development of systems by incremental assembling of components.