As already said by Simon in his comment, this is a very vast topic.
Let us stick for simplicity to the case of 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 case of 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 such phenomena can be regarded as a particular case of a more general situation that he called the Murphy's law for moduli spaces.
In recent years, many people studied possible compactifications of $\mathfrak{M}_{\chi, \, K^2}$. For instance, Kollar and Shepherd Barron proposed to add surfaces with semi-log canonical singularities and ample $K_S$, that they called stable surfaces by analogy with stable curves. Using this construction, Alexeev and Pardini where for instance able to provide explicit compactifications of the moduli space of Burniat and Campedelli surfaces.
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.
J. Kollar and N. I- Shepherd-Barron: Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299-338.
V. Alexeev, R. Pardini: Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, arXiv:0901.4431.