Abstract: As of today, there exists no standard language for querying Linked Data on the Web, where navigation across distributed data sources is a key feature. A natural candidate seems to be SPARQL, which recently has been enhanced with navigational capabilities thanks to the introduction of property paths (PPs). However, the semantics of SPARQL restricts the scope of navigation via PPs to single RDF graphs. This restriction limits the applicability of PPs on the Web. To fill this gap, in this paper we provide formal foundations for evaluating PPs on the Web, thus contributing to the definition of a query language for Linked Data. In particular, we introduce a query semantics for PPs that couples navigation at the data level with navigation on the Web graph. Given this semantics we find that for some PP-based SPARQL queries a complete evaluation on the Web is not feasible. To enable systems to identify queries that can be evaluated completely, we establish a decidable syntactic property of such queries.