In the study of multiplicity free actions of reductive Lie algebras, one is naturally lead to invariant polynomial coefficients differential operators. The abstract and concrete Capelli problems of Howe and Umeda ask for the realization of invariant differential operators by means of the centre of the universal enveloping algebra. We will review this classical theory and describe our new results for Lie superalgebras. This talk is based on joint work with Siddhartha Sahi.