Abstract: Let $G \times X \rightarrow X$ be an action of the algebraic group $G$ on the affine, algebraic variety $X$. There are two quite different notions of quotient associated with this situation. If we embrace the conventional approach, we just accept the object $Y =$ Spec$(k[X]^G)$, along with the natural map $\pi : X \rightarrow Y$, as the inevitable thing to study. If $G$ is reductive then this "quotient" is a variety and it has the anticipated universal property, even if $Y$ is not an orbit space. Similar results hold if $X$ is a projective variety. Many important moduli spaces have been constructed using this approach. But maybe there is another approach, where the emphasis is on orbits of maximal dimension rather than on closed orbits. In this scenario we consider sufficiently small open, $G$-subsets $U$ of $X$ such that each $G \times U \rightarrow U$ has as many desirable properties as the situation will tolerate. If we define $U/G$ by the equation $k[U/G] = k[U]^G$, then we can ask for the following. (1) $k[U/G]$ is finitely generated. (2) $k[U/G]$ is a regular ring. (3) $\pi : U \rightarrow U/G$ is surjective. (4) $\pi : U \rightarrow U/G$ separates orbits of maximal dimension. (5) $\pi : U \rightarrow U/G$ has no exceptional divisors. (6) $\pi : U \rightarrow U/G$ is flat. We do this so as to discard only a small portion of $X$. We then try to glue all these $U/G$'s together to get a separated quotient variety, without the help of semi-invariants.