Abstract: Let $G$ be an affine algebraic group and let $X$ be an irreducible, affine variety. Assume that $G$ acts on $X$ via $G \times X \to X$. The action is called stable if there exists a nonempty, open subset $U\subseteq X$ consisting entirely of closed $G$-orbits. The action is called observable if for any proper, $G$-invariant, closed subset $Y\subseteq X$ there is a nonzero invariant function $f\in k[X]^G$ such that $f|_Y = 0$. It is easy to prove that "observable implies stable" but the two notions are not the same for general groups. We discuss a useful geometric characterization of observability. We then discuss some of the following questions and illustrate them with the appropriate examples.

(1) When is the action $H \times G\to G$, by left translation, observable?
(2) Does the characterization simplify if G is unipotent? solvable? reductive?
(3) What happens if $X$ is factorial? reducible?
(4) Is $int : G\times G\to G$, $(g,x)\mapsto gxg^{-1}$, always observable?
(5) Can we generalize to the case where $X$ is projective and $G \times X \to X$ is linearizable?