Query execution over the Web of Linked Data has attracted much attention recently. A particularly interesting approach is link traversal based query execution which proposes to integrate the traversal of data links into the construction of query results. Hence -in contrast to traditional query execution paradigms- this approach does not assume a fixed set of relevant data sources beforehand; instead, it discovers data on the fly and, thus, enables applications to tap the full potential of the Web. While several authors study possibilities to implement the idea of link traversal based query execution and to optimize query execution in this context, no work exists that discusses the theoretical foundations of the approach in general. Our paper fills this gap. We introduce a well-defined semantics for queries that may be executed using the link traversal based approach. Based on this semantics we formally analyze properties of such queries. In particular, we study the computability of queries as well as the implications of querying a potentially infinite Web of Linked Data. Our results show that query computation in general is not guaranteed to terminate and that for any given query it is undecidable whether the execution terminates. Furthermore, we define an abstract execution model that captures the integration of link traversal into the query execution process. Based on this model we prove the soundness and completeness of link traversal based query execution and analyze an existing implementation approach..