For questions about the geometry, complex-analytic, and group-theoretic properties of the Möbius transformations (linear fractional transformations) $z \mapsto \frac{az + b}{cz + d}$ of the complex plane, which can be identified with the group $PGL(2, \mathbb{C})$, or certain subgroups thereof.
Möbius transformations, sometimes called linear fractional transformations, are invertible meromorphic transformations of the complex plane (or holomorphic transformations of the Riemann sphere) of the form $$z \mapsto \frac{a z + b}{c z + d}, \qquad ad - bc \neq 0.$$ Two matrices $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ determine the same transformation iff one is a complex multiple of the other, and so the group of Möbius transformations can be identified with the complex Lie group $PSL(2, \mathbb{C})$.
The subgroup $PSL(2, \mathbb{R})$ of transformations for which $a, b, c, d$ are real and $ad - bc > 0$ preserves the upper half-plane $\{\Im z > 0\}$ and in particular the defining metric of the Poincare upper half-plane model of the hyperbolic plane.
Möbius transformations preserve the cross-ratio $$\frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)},$$ a classical projective invariant of a quadruple $(z_1, z_2, z_3, z_4)$ of points.
Sometimes the term Möbius transformation is used to refer just to transformations of the particular form $$ z \mapsto \frac{z - a}{1 - \bar{a} z}, \qquad a \in \mathbb{D}, $$ which in particular preserves the unit disk $\mathbb{D}$.
Sources: Wikipedia, Wolfram Mathworld
