Holomorphic motions were introduced by Mane, Sad and Sullivan in the 1980's motivated by applications in holomorphic dynamics. They have since then been applied in various other areas such as Kleinian groups, Teichm\"{u}ller theory and most notably quasiconformal mappings. In this talk, I will discuss recent results on the variation of various notions of dimension (Minkowski, Packing, Hausdorff) under holomorphic motions. The results involve a new class of functions called Inf-harmonic.