Compactifying Moduli Spaces for Abelian Varieties This volume presents the construction of canonical modular compactifications of moduli spaces for polarized Abelian varieties (possibly with level structure) building on the earlier work of Alexeev Nakamura and Namikawa. This provides a different approach to compactifying these spaces than the more classical approach using toroical embeddings which are not canonical. There are two main new contributions in this monograph: (1) The introduction of logarithmic geometry as understood by Fontaine Illusie and Kato to the study of degenerating Abelian varieties; and (2) the construction of canonical compactifications for moduli spaces with higher degree polarizations based on stack theoretic techniques and a study of the theta group. A Brief Primer on Algebraic Stacks Preliminaries Moduli of Broken Toric Varieties Moduli of Principally Polarized Abelian Varieties Moduli of Abelian Varieties with Higher Degree Polarizations Level Structure.