the uniformization theorem
There are infinitely many Riemann surfaces, of every genus and shape — it seems hopeless to organize them. The uniformization theorem performs an astonishing reduction: up to the right notion of sameness, there are only THREE simply connected Riemann surfaces in the entire universe, and every other surface is a quotient of one of them. It is the complex-analytic analogue of saying every constant-curvature geometry is spherical, flat, or hyperbolic.
Precisely, the uniformization theorem states that every simply connected Riemann surface is biholomorphic to exactly one of three model spaces: the Riemann sphere (the compact, positively curved model), the complex plane C (the flat model), or the open unit disk D, equivalently the upper half-plane (the hyperbolic, negatively curved model). Consequently, for ANY Riemann surface X, pass to its universal cover X-tilde, which is simply connected, so X-tilde is one of the three; then X = X-tilde / Gamma, where Gamma is a group of deck transformations acting freely and properly by biholomorphisms (Mobius transformations of the model). So every Riemann surface is a model space modulo a discrete group of automorphisms.
Why it matters: this single theorem classifies and uniformizes everything. The trichotomy lines up with genus: the sphere itself (g = 0) is its own cover; surfaces covered by C are exactly the torus C/L and a few exceptions (g = 1, flat geometry); and EVERY surface of genus g >= 2 is covered by the disk, hence carries a canonical complete hyperbolic metric of curvature -1. That is the deep reason 'most' Riemann surfaces are intrinsically hyperbolic — the link between complex analysis and hyperbolic geometry. Honesty points: the theorem identifies the universal cover up to biholomorphism, but the SURFACE still depends on the group Gamma; many non-isomorphic surfaces share the disk as cover (this is why moduli space is interesting, not a point). And it is a hard theorem (Koebe, Poincare), strictly stronger than the Riemann mapping theorem, which only handles simply connected proper open subsets of C.
A genus-2 surface has the disk as universal cover and is realized as D / Gamma, where Gamma is a discrete group of disk automorphisms (a Fuchsian group). Concretely Gamma is the fundamental group acting by Mobius maps preserving the disk; the surface inherits the hyperbolic metric of the disk, of constant curvature -1, and a hyperbolic area equal to 2pi(2g - 2) = 4pi by Gauss-Bonnet.
Genus >= 2 surfaces are disk quotients D/Gamma and thus carry a canonical curvature -1 hyperbolic metric.
Uniformization fixes the universal cover (one of three), but NOT the surface — that depends on the deck group Gamma, so countless inequivalent surfaces share the disk as cover. It is strictly stronger than the Riemann mapping theorem and is what makes genus >= 2 surfaces canonically hyperbolic.