The representation theories of symmetric groups and of general linear groups are linked through Schur-Weyl duality. In 1937, Brauer asked the following question: which algebra has to replace the group algebra of the symmetric group in this situation if one replaces the general linear group by its orthogonal or symplectic subgroup? As an answer he defined the Brauer algebra. We will discuss this, and also see how a theory of Young modules leads to another Schur-Weyl duality for Brauer algebras. This is joint work with Robert Hartmann, Anne Henke and Steffen Koenig.