In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 168 × 2 = 336 automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to PSL(2, 7), the second-smallest non-abelian simple group after the alternating group A₅. The quartic was first described in (Klein 1878b).

Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, Fermat's Last Theorem, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see (Levy 1999) for a survey of properties.

Originally, the "Klein quartic" referred specifically to the subset of the complex projective plane P²(C) defined by an algebraic equation. This has a specific Riemannian metric (that makes it a minimal surface in P²(C)), under which its Gaussian curvature is not constant. But more commonly (as in this article) it is now thought of as any Riemann surface that is conformally equivalent to this algebraic curve, and especially the one that is a quotient of the hyperbolic plane H² by a certain cocompact group G that acts freely on H² by isometries. This gives the Klein quartic a Riemannian metric of constant curvature −1 that it inherits from H². This set of conformally equivalent Riemannian surfaces is precisely the same as all compact Riemannian surfaces of genus 3 whose conformal automorphism group is isomorphic to the unique simple group of order 168. This group is also known as PSL(2, 7), and also as the isomorphic group PSL(3, 2). By covering space theory, the group G mentioned above is isomorphic to the fundamental group of the compact surface of genus 3.

It is important to distinguish two different forms of the quartic. The closed quartic is what is generally meant in geometry; topologically it has genus 3 and is a compact space. The open or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically these punctures are cusps. The open quartic may be obtained (topologically) from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below. The open and closed quartics have different metrics, though they are both hyperbolic and complete – geometrically, the cusps are "points at infinity", not holes, hence the open quartic is still complete.

The Klein quartic can be viewed as a projective algebraic curve over the complex numbers C, defined by the following quartic equation in homogeneous coordinates [x:y:z] on P²(C):

The locus of this equation in P²(C) is the original Riemannian surface that Klein described.

The compact Klein quartic can be constructed as the quotient of the hyperbolic plane by the action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the ideal ${\displaystyle I=\langle \eta -2\rangle }$ in the ring of algebraic integers Z(η) of the field Q(η) where η = 2 cos(2π/7). Note the identity

exhibiting 2 – η as a prime factor of 7 in the ring of algebraic integers.