TITLE: What are some examples of $\text{Isom}(M)$ and $\text{Conf}(M)$?
QUESTION [9 upvotes]: Edit:
Since I did not quite get the responses I would have liked when I asked this question four months ago, let me reformulate it slightly:

What are some examples of $\text{Isom}(M)$ and $\text{Conf}(M)$?

For example, Aaron mentions in his (very helpful) answer that $\text{Isom}(\mathbb{S}^n) \cong O(n+1)$.  What about hyperbolic n-space?  Or the n-torus?  The more examples the better.
For precision, I am hoping that we can express $\text{Isom}(M)$ or $\text{Conf}(M)$ as some sort of "recognizable" Lie group, by which I mean a product, quotient, and/or connected sum of linear groups, euclidean spaces, or spheres.

Original Question: (What are some examples of automorphism groups of manifolds which turn out to be Lie groups?)
I recently read that the group of diffeomorphisms of a smooth manifold which preserve some sort of geometric structure (e.g. Riemannian structure, conformal structure, etc.) frequently turn out to be a Lie group.  What are some examples of this?
I've read about the euclidean group $E(n)$ which are the isometries of $\mathbb{R}^n$, and also about the conformal automorphisms of the complex plane, upper half plane, and unit disc.  What others are there?  Do the groups Diff(M), Iso(M), etc. frequently turn out to be something recognizable?  What are the isometry groups or conformal groups of n-spheres, say, or some common 2-manifolds?

REPLY [3 votes]: $\text{Isom}(\mathbb{S}^n) \cong O(n+1)$
$\text{Isom}(\mathbb{R}^n) \cong E(n)$
$\text{Isom}(\mathbb{H}^n) \cong O(n,1)$
$\text{Conf}(\mathbb{S}^2) \cong \text{PGL}(2,\mathbb{C}) \cong \text{PSL}(2,\mathbb{C})$
$\text{Conf}(U) \cong \text{PSL}(2,\mathbb{C})$
Sources: "Riemannian Manifolds: An Introduction to Curvature" (Lee) (Chapter 3), and my own understanding of Wikipedia (and my complex analysis lecture notes).