I'm dealing with Lagrange multipliers and one way to determine whether a maximum or minimum exists is to check whether the domain, given by the constraint, is compact. For example, $x^2 + y^2 = 1$, the unit circle, is compact so given the function is continuous, the image would definitely have a max and a min.
However, are there quick ways or tricks to determine whether the domain is compact? For example, is the paraboloid $x^2+y^2-z$ compact?
Yes. In $\mathbb R^n$ being compact is equivalent to being closed and bounded. This is the Heine Borel Theorem. closedness means intuitely that the set contains all of its limit points (so something like $\{x,y \mid x^2+y^2<1\}$ will not work, since $(1,0)$ is not in the set, but there is some sequence of points approaching it.) In your case, when the domain is given by some equation. If the domain is the zero set of a continuous function ($x^2+y^2-1$), it is closed. If it is also bounded, then it is compact.
The collection $x^2+y^2-z$ is not compact, since it is unbounded (look at the picture.)