Weak Decays of $B$ Mesons and Lattice QCD

After a pedagogical introduction to the calculation of weak matrix elements on the lattice, I review some of the lattice's most recent predictions concerning the weak decays of $B$-mesons. Amongst the topics covered are the determinations of the leptonic decay constant $f_B$, of the form factors relevant for semi-leptonic $B\to D(D^*)\ell\bar\nu$ decays and of the hadronic matrix element which describes the rare decay $B\to K^*\gamma$. Emphasis is placed on the results of the UKQCD Collaboration. Corrections to the heavy-quark limit are discussed extensively.