Abstract: Subordination is the operation that evaluates a Lévy process at a subordinator, giving rise to a path-wise construction of a "time-changed'' process. In probability semigroups, subordination was applied to create the variance gamma process, which is prominently used in financial modelling. However, subordination may not produce a Lévy process unless the subordinate has independent components or the subordinate has indistinguishable components. A new operation known as weak subordination is introduced that always produces a Lévy process by assigning the distribution of the subordinate conditional on the value of the subordinate, which matches traditional subordination in law in the cases above. Weak subordination is applied to extend the class of variance-generalised gamma convolutions and to construct the weak variance-alpha-gamma process. The latter process exhibits a wider range of dependence than using traditional subordination. This is joint work with Prof Dilip B Madan (University of Maryland, USA) and Dr Kevin W Lu (Australian National University).