Abstract: First I will give a short introduction to Nichols algebras of Yetter-Drinfeld modules over arbitrary Hopf algebras $H$. To understand Nichols algebras over group algebras is the first fundamental step to classify pointed Hopf algebras. The plus part of the Drinfeld-Jimbo quantum group of a semisimple Lie algebra is an important example of a Nichols algebra where $H$ is the group algebra of a free abelian group of finite rank. Then I will describe axioms for root systems and Weyl groupoids extending the familiar notions of root systems anmd Weyl groups of semisimple Lie algebras. This is a reformulation of the axioms given by I. Heckenberger and H. Yamana. Finally I will discuss the root system and the Weyl groupoid of a Nichols algebra of a semisimple Yetter-Drinfeld module introduced in recent joint work with I. Heckenberger and based on recent joint work with N. Andruskiewitsch and I. Heckenberger. There are applications to the classification of finite-dimensional pointed Hopf algebras over the symmetric group $S_3, S_4$ and the dihedral groups.