We discuss a number of algorithmic problems, and possible solutions, for algebraic groups in characteristic 0. We will talk about some basic algorithms, that is, how to specify an algebraic group, computing the dimension, the Lie algebra, centralizers and normalizers, the closure of an orbit. Secondly we will look at the problem to compute the Zariski closure of a finitely generated matrix group. This also involves the related question of how to compute the smallest algebraic Lie algebra containing a given Lie algebra. A third topic is the problem how to find generators of arithmetic groups. These arise as the set of integral points of an algebraic group defined over Q. A famous theorem by Borel and Harish-Chandra asserts that these groups are finitely generated. But it remains a very hard problem to find a finite generating set. Algorithms exist for some classes of algebraic groups