Many articles by Bridgeland.
http://ncatlab.org/nlab/show/reconstruction+theorem
Orlov paper on representability of eqiovalences by objects on the product
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.
http://front.math.ucdavis.edu/0912.4040 Title: Derived invariance of the number of holomorphic 1-forms and vector fields. Authors: Mihnea Popa, Christian Schnell. Abstract: We prove that smooth projective varieties with equivalent derived categories have isogenous (and sometimes isomorphic) Picard varieties. In particular their irregularity and number of independent vector fields are the same. This is turn implies, in combination with the invariance of Hochschild homology, that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.
nLab page on Derived categories in algebraic geometry