In the present contribution we look at the legacy of Hilbert’s programme in some re-cent developments in mathematics. Hilbert’s ideas have seen new life in generalised andrelativised forms by the hands of proof theorists and have been a source of motivationfor the so–called reverse mathematics programme initiated by H. Friedman and S. Simp-son. More recently Hilbert’s programme has inspired T. Coquand and H. Lombardi toundertake a new approach to constructive algebra in which strong emphasis is laid onthe use of finite methods. The main aim is to eliminate the ideal objects and in so doingobtain more elementary and informative proofs. We survey some work in commutativealgebra—mainly about and around the Zariski spectrum and the Krull dimension of acommutative ring—which witnesses the feasibility of such a revised Hilbert programme.