  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.
