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, Teichmüller theory and most notably quasiconformal mappings. In this talk, I will discuss recent results on the variation of several quantities under holomorphic motions, such as logarithmic capacity, (continuous) analytic capacity, Minkowski/Packing/Hausdorff dimension, and Hausdorff measure. This is based on joint works with Wen-Hui Ai, Aidan Furher and Thomas Ransford.