Summary: Global types are formal specifications that describe communication protocols in terms of their global interactions. We present a new, streamlined language of global types equipped with a trace-based semantics and whose features and restrictions are semantically justified. The multi-party sessions obtained projecting our global types enjoy a liveness property in addition to the traditional progress and are shown to be sound and complete with respect to the set of traces of the originating global type. Our notion of completeness is less demanding than the classical ones, allowing a multi-party session to leave out redundant traces from an underspecified global type. In addition to the technical content, we discuss some limitations of our language of global types and provide an extensive comparison with related specification languages adopted in different communities.