I will discuss two pieces of work. In the first we present theoretical results related to preconditioning Markov chain Monte Carlo sampling algorithms. Preconditioning is a widely used technique that is known empirically to improve the mixing of Markov chain algorithms, but little has been said theoretically about it. I will present some recent work establishing some positive and negative theoretical results. I will then discuss some methodological work devising new preconditioners. The standard options to choose from in common software packages are 'diagonal' or 'dense'. We will present a new alternative option that seeks to improve upon diagonal preconditioning whilst also being less computationally expensive than the quadratic cost required for the dense option. Both projects are joint work with my former PhD student Max Hird, now a PDRA at University of Waterloo. The first is associated with this paper: https://www.jmlr.org/papers/v26/23-1633.html.