As already said by Simon in his comment, this is a very vast topic.
Let us stick for simplicity to smooth surfaces $S$ of general type.
The existence of a quasiprojective coarse moduli space $\mathfrak{M}$ for such surfaces modulo birational equivalence was proven by Gieseker using GIT theory and the birationality of $5$-canonical map proven by Bombieri. It follows that for fixed values of $\chi(\mathcal{O}_S)$ and $K_S^2$ the space $\mathfrak{M}_{\chi, \, K^2}$ has a finite number of irreducible components.
As in the case of $\mathcal{M}_g$, locally in a neighborhood of a point $[S]$ the moduli space $\mathfrak{M}$ is given by a quotient of the base $\textrm{Def}(S)$ of the Kuranishi family of $S$ by the finite group $\textrm{Aut}(S)$. However, in contrast with the casef curves, often patologies arise, for instence $\textrm{Def}(S)$ can be non reduced. This was first explained by Catanese in the case when $K_S$ is not ample, then Vakil later proved that this kind of phenomena can be regarded as a particular case of a more general situation that he called the Murphy's law for moduli spaces.
Here is a (short) bibliography on the subject:
D. Gieseker: Global moduli for surfaces of general tipe, Invent. math 43 (1977), 233-282.
F. Catanese: On the moduli spaces of surfaces of general type, J. Differential geometry 19 (1984), 483-515.
F. Catanese: Everywhere non riduced moduli spaces, Invent. math. 98 (1989), 293-310.
R: Vakil: Murphy's Law in algebraic geometry: Badly-behaved deformation spaces, Invent. Math. 164 (2006), 569-590.