the moduli space of curves
We saw that all genus-g surfaces are topologically the same, yet they form a whole family of inequivalent Riemann surfaces. The moduli space of curves is the geometric object whose POINTS are these isomorphism classes: one point for each genus-g Riemann surface up to biholomorphism. Instead of studying surfaces one at a time, you study the space of all of them at once — its dimension, its shape, how surfaces degenerate at its boundary.
Precisely, M_g denotes the moduli space of smooth compact Riemann surfaces (equivalently smooth projective curves over C) of genus g, where two surfaces give the same point exactly when they are biholomorphic. For g >= 2 it is a connected complex space (in fact a quasi-projective variety, but with mild orbifold singularities coming from curves with automorphisms) of complex dimension 3g - 3. The low genera are special: M_0 is a single point (every genus-0 surface is the sphere), and M_1 is one-dimensional, parametrized by the j-invariant (genus-1 curves with a marked point, the complex plane of j-values). The boundary, where curves acquire nodes and pinch, is added to form the Deligne-Mumford compactification M_g-bar, whose study organizes how families of curves degenerate.
Why it matters: moduli space is where the classification problem lives and where modern algebraic geometry, topology, and even physics meet. Its dimension 3g - 3 (the number of complex parameters needed to specify a genus-g surface) is a fundamental count; its cohomology and intersection theory (Mumford classes, Witten's conjecture proved by Kontsevich) connect to string theory and integrable systems. Honesty caveats — several. M_g is a COARSE moduli space and has orbifold/stack structure because some curves have automorphisms; the clean object is the moduli STACK. It is NOT compact (curves can degenerate), which is why one passes to the Deligne-Mumford compactification. And this is distinct from Teichmuller space, which is the simply connected cover of M_g obtained by rigidifying with a marking; M_g is the quotient of Teichmuller space by the mapping class group, and that genuinely separate topic belongs to Geometric Structures and Teichmuller theory, not here.
For genus 1, the moduli space M_1 is parametrized by a single number, the j-invariant: two elliptic curves C/L and C/L' are isomorphic exactly when j(L) = j(L'), and j ranges over all of C. So the family of all tori, infinite as it looks, is organized by one complex coordinate — dimension 3g - 3 = 0 plus the marked-point correction gives the expected 1-dimensional picture for elliptic curves.
Every elliptic curve is pinned by its j-invariant; M_1 is the j-line, one complex parameter.
M_g is a COARSE (orbifold/stack) moduli space — automorphisms create singularities, and it is non-compact, so one uses the Deligne-Mumford compactification. It is NOT Teichmuller space: M_g is the quotient of Teichmuller space by the mapping class group, a separate topic.