A locking protocol is a set of rules governing the manner in which the database entities may be accessed. Such a protocol usually employs several kinds of locks. Most of the previous work in this area has assumed that once a transaction acquires a particular kind of lock on a data item it is not allowed to convert this lock to another kind. In this paper we perform a systematic study of the consequences of allowing lock conversions in non-two-phase locking protocols, and show how this leads to increased concurrency and affects deadlock-freedom. The non-two-phase protocols that we study are the very general guard protocols defined for databases in which a directed acyclic graph structure can be superimposed on the data items. We present very natural generalizations of these protocols, including correctness proofs, and develop deadlock removal methods.