In algebraic geometry, the existence and geometry of quotient schemes is a delicate issue.  Even when quotients exist, they may not reflect enough properties of the original group action to be useful.  The machinery of geometric invariant theory is one prescription for identifying open subsets of the original scheme that admit useful quotients, but it can be shown that there are, in general, other open sets that also admit well-behaved quotients.  In this talk, we examine particular actions of diagonalizable groups on affine space and illustrate the wide variety of properties that quotients arising from this action can have.