Braid group actions on vector space tensor powers are used to define a large class of associative algebras, termed Nichols algebras. These Nichols algebras underlie the structure of many Hopf algebras, including the small quantum groups, and are essential in classification theorems for some types of finite dimensional Hopf algebras. The structure of Nichols algebras, e.g. PBW-type bases in some settings, is governed by braiding. This facilitates homological techniques for understanding representations. We will introduce cohomology of finite dimensional Hopf algebras, and state an open finite generation conjecture. We will describe some recent results for Hopf algebras arising from Nichols algebras.