We consider a reductive algebraic group G defined over the reals which acts linearly on a vector space V . It is classically known that the algebra of invariant polynomials is finitely generated, which allows one to construct an algebraic orbit map V ¿ V/G. After introducing the setup and recalling some important results by Procesi-Schwarz, Brocker and others, we will take a closer look at G-invariant polynomials which are positive or sums of squares.