The Siegel upper half-space H_g serves as a natural generalization of the upper half-plane. Indeed, just as M_{1,1} = A_1 is the quotient H/SL_2(Z), one can show that A_g, the space of principally polarized abelian varieties of dimension g, is isomorphic to H_g/Sp_{2g}(Z). More generally, one can consider spaces H_g/Gamma for any arithmetic subgroup Gamma in Sp_{2g}(Q). Unlike M_g, which admits the canonical Deligne-Mumford compactification, these spaces admit many natural compactifications with different geometric properties. In this talk, we introduce H_g/Gamma, the Siegel modular varieties, and discuss various compactifications such as the Baily–Borel compactification and toroidal compactifications. We then focus on the specific case of g = 2 to showcase the beautiful arithmetic that occurs even in this small case.