I will report on recent joint work with I. Heckenberger. We are studying systematically the Nichols algebra (or quantum symmetric algebra) of a Yetter-Drinfeld module over any Hopf algebra (with bijective antipode) which is a finite direct sum of finite-dimensional irreducible Yetter-Drinfeld modules. In this general context we define reflection maps in joint work with N. Andruskiewistch. In the special case of the quantum groups in Lusztig's book these maps are essentially the restriction of the Lusztig automorphisms to the plus part of the quantum group. Under mild assumptions we associate a generalized root system (in the sense of Heckenberger and Yamane) and a Weyl groupoid to the Nichols algebra. Using these invariants it is possible to decide when the Nichols algebra is finite-dimensional. We obtain a coproduct formula which seems to be new even for the classical quantum groups. Then we describe the right coideal subalgebras of the Nichols algebra by words in the Weyl groupoid. As a special case we obtain a proof of a recent conjecture of Kharchenko which says that the number of right coideal subalgebras of the plus part of the quantum group of a semisimple Lie algebra is the order of the Weyl group.