One very widely used criterion in the theory of Markov chains states that if a Markov operator has the strong Feller property and is topologically irreducible, then it can have at most one invariant measure. While this criterion is very useful in finite-dimensional situations, it fails for many infinite-dimensional problems. In this talk, we will present two different generalisations of the strong Feller property that can be applied to a much larger class of problems. These include semilinear parabolic stochastic PDEs, stochastic delay equations, and diffusions driven by fractional noise.