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The first part of this paper is a refinement of Winkelmann's work oninvariant rings and quotients of algebraic groups actions on affine varieties,where we take a more geometric point of view. We show that the (algebraic)quotient $X/\!/\!G$ given by the possibly not finitely generated ring ofinvariants is "almost" an algebraic variety, and that the quotient morphism$\pi\colon X \to X/\!/\! G$ has a number of nice properties. One of the maindifficulties comes from the fact that the quotient morphism is not necessarilysurjective.  These general results are then refined for actions of the additive group$\mathbb{G}_a$, where we can say much more. We get a rather explicitdescription of the so-called plinth variety and of the separating variety,which measures how much orbits are separated by invariants. The most completeresults are obtained for representations. We also give a complete and detailedanalysis of Roberts' famous example of a an action of $\mathbb{G}_a$ on7-dimensional affine space with a non-finitely generated ring of invariants.

Invariants and Separating Morphisms for Algebraic Group Actions