  The theory of quasi-Lie systems, i.e. systems of first order ordinary
differential equations which can be related via a generalised flow to Lie
systems, is extended to systems of partial differential equations and its
applications to obtaining $t$-dependent superposition rules and integrability
conditions are analysed. We develop a procedure of constructing quasi-Lie
systems through a generalisation to PDEs of the so-called theory of quasi-Lie
schemes. Our techniques are illustrated with the analysis of
Wess-Zumino-Novikov-Witten models, generalised Abel differential equations,
Baecklund transformations, as well as other differential equations of physical
and mathematical relevance.
