In the Fastest Mixing Markov Chain problem, we are given a graph $G = (V, E)$ and desire the discrete-time Markov chain with smallest mixing time $τ$ subject to having equilibrium distribution uniform on $V$ and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time $τ_\textsf{RW}$ of the lazy random walk on $G$ is characterised by the edge conductance $Φ$ of $G$ via Cheeger's inequality: $Φ^{-1} \lesssim τ_\textsf{RW} \lesssim Φ^{-2} \log |V|$. Analogously, we characterise the fastest mixing time $τ^\star$ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance $Ψ$ of $G$: $Ψ^{-1} \lesssim τ^\star \lesssim Ψ^{-2} (\log |V|)^2$. This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on $G$ with equilibrium distribution which need not be uniform, but rather only $\varepsilon$-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time $τ\lesssim \varepsilon^{-1} (\operatorname{diam} G)^2 \log |V|$. Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.