Gross-Hacking-Keel-Kontsevich proposed a construction of Landau-Ginzburg models for cluster varieties. This class includes many examples of interest in representation theory, the simplest of which are perhaps the Grassmannians Gr(k,n). I will report on recent progress in establishing homological mirror symmetry in this case, building on work of Marsh-Rietsch and Rietsch-Williams. In particular, I will highlight two results: a complete proof in the case n prime, where the Fukaya category is generated by the monotone Lagrangian torus fiber of an integrable system introduced by Guillemin-Sternberg in the '80s; the existence of exotic monotone Lagrangian tori corresponding to different charts of the mirror, obtained by bootstrapping from the initial torus through a mutation procedure which is compatible with the general picture of mirror symmetry for Fano varieties put forth by Coates, Corti, Galkin, Golyshev and others.