Complex affine $n$-space $C^n$, the basic object of algebraic geometry, offers a number of exciting and striking problems. The most famous one is the still unsolved Jacobian Conjecture. Others are the Cancellation Problem (Does $Y \times C^k \simeq C^{n+k}$ imply that $Y \simeq C^n$ ?), the Linearization Problem (Is every automorphism of $C^n$ of finite order conjugate to a linear automorphism ?), or the Embedding Problem (Are there other embeddings of $C^{n-1}$ into $C^n$ than the standard ones ?). It turns out that these questions and several others are intimately related and have very interesting connections with problems arising from algebraic group actions and orbit spaces. We give a survey on these problems and discuss some recent progress and examples.