Weak decays of beautiful mesons can be used to probe the validity of our best model of particle physics: the Standard Model. In this thesis, I use specialised techniques to overcome practical challenges associated with the large mass of valence $b$ quarks to study a selection of decays of $B$ mesons with Lattice Quantum Chromodynamics. After introducing these modern methods, I report on my progress on calculating the vector current form factors for $Bc\to Ds$. This decay is allowed at the one-loop level and so my vector current form factors will be used, along with future analysis of the tensor form factor, to describe the short-distance physics of this transition.

I then present a comprehensive calculation of the $Bc\to Bs(d)$ vector current form factors based on a non-relativistic field theory approach for the bottom quark. I conclude by comparing my results with another calculation that uses an alternative approach for the heavy valence quark, an important test of our best lattice techniques. Combining data from this study with my calculation, I give the first prediction of the decay rates $\Gamma (Bc+\to Bs0\ell ―\nu \ell )$ and $\Gamma (Bc+\to B0\ell ―\nu \ell )$ from lattice QCD.

Next, exploratory work on the long-distance contributions of charmonia on the flavour-changing neutral current process $B\to K$ is discussed. Difficulties with intermediate states that lead to divergences in the corresponding correlation function are addressed by again using the non-relativistic approach for the $b$ quark.

Finally, I digress from weak decays to explore applications of the Sherman-Morrison-Woodbury formula within the framework of lattice QCD. I find computational efficiencies in determining the inverse and the determinant of quark matrices. Potential applications are discussed.