Many things in folder AG/Moduli
http://mathoverflow.net/questions/13868/proper-definition-of-a-moduli-problem
arXiv:1004.3259 Open problems (for AGNES) from arXiv Front: math.AG by R. Pandharipande The article contains a few questions and speculations related to the moduli spaces of curves, K3 surfaces, maps, and sheaves presented in the problem session of the AGNES conference in Amherst (April 2010).
Harris: An introduction to the moduli space of curves
Harris and M?: Moduli of curves
Arithmetic moduli of elliptic curves, by Katz and Mazur
http://nlab.mathforge.org/nlab/show/moduli+space
What about good refs for moduli of vector bundles?
Two books to be published by CUP in 2009: Bradlow et al: Moduli spaces of VBs, Huybrechts and Lehn: The Geometry of Moduli spaces of sheaves.
Ramanan et al ed, Moduli spaces and VBs, LMS lecture note series
In general, what kind of objects can be described by moduli spaces? Curves and VBs, maybe cycles/subvarieties. What else??? Metrics? t-structures (Bridgeland)?
Bridgeland says in the intro of Fourier-Mukai transforms for K3 and elliptic fibrations, that we know quite a lot about moduli of stable VBs on projective surfaces, but not so much about VBs on higher-dimensional varieties. The paper tries to do something about this in certain cases I think.
Blanc, Dwyer, Goerss: The realization space of a Pi-algebra. Studies the moduli space of spaces with a fixed structured homotopy groups. It is defined as the classifying space of the category whose objects are such top spaces, and morphisms are weak equivalences. Description in terms of a tower. These methods might apply to other moduli problems.
There are various papers by Olsson et al, not downloaded.
arXiv:1008.0621 Moduli of varieties of general type from arXiv Front: math.AG by János Kollár This is a survey paper discussing the moduli problem for varieties of general type.
About moduli for higher-dimensional varieties: Toen and Anel prove that the number of iso classes of smooth projective complex alg vars with the same derived cat must be countable. File Toen web publ dgcat-alg.pdf. They use some kind of (nonalgebraic) stack which in some sense is a moduli space for smooth projective varieties.
arXiv:0908.1938 Stability phenomena in the topology of moduli spaces from arXiv Front: math.AT by Ralph L. Cohen The recent proof by Madsen and Weiss of Mumford’s conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where “stability phenomena” occur in their topologies. Such stability theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical stability theorems such as the Freudenthal suspension theorem, Bott periodicity, and Whitney’s embedding theorems. We then discuss more modern examples such as those involving configuration spaces of points in manifolds, holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the stable topology of general linear groups, and pseudoisotopies of manifolds. We then discuss the stability theorems regarding the moduli spaces of Riemann surfaces: Harer’s stability theorem on the cohomology of moduli space, and the Madsen-Weiss theorem, which proves a generalization of Mumford’s conjecture. We also describe Galatius’s recent theorem on the stable cohomology of automorphisms of free groups. We end by speculating on the existence of general conditions in which one might expect these stability phenomena to occur.
nLab page on Moduli spaces