A conformal transformation is a diffeomorphism that preserves a conformal structure (a conformal map). The group of conformal transformations is the conformal group.
A bijective conformal transformation of the Riemann sphere (to itself) is also called a Möbius transformation.
For $p + q \geq 3$ then the group of conformal transformations of $\mathbb{R}^{p,q}$ is $\simeq O(p+1,q+1)$
Wolfgang Kühnel, Hans-Bert Rademacher, Liouville’s theorem in conformal geometry (pdf)
Isadore Singer, Shlomo Sternberg, The infinite groups of Lie and Cartan. J. Anal. Math. 15, 1-114 (1965)
briefly reviewed in
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