Abstract: One of the main themes of invariant theory is to relate the $G$-invariant regular functions, of a regular action $G\times X\to X$, to some suitable quotient morphism $\pi : X\to Y$. However, there are examples to show that the naive attempt $X\mapsto k[X]^G$ does not lead directly to any appealing conclusion. Indeed, $k[X]^G$ may not be finitely generated, or it may not be "large" enough to separate the $G$-orbits of $G\times X\to X$, even generically.

The purpose of this talk is to discuss some basic results of "quasi-invariant theory". The main ideas here have their roots in the work of Hilbert, Zariski, Nagata, and Rosenlicht. Our major purpose is to assess the influence of quasi-invariant rational functions and $G$-invariant divisors on the problem of constructing a useful quotient object of a regular action $G\times X\to X$.