HIV assembly and replication proceed through the formation of morphologically distinct immature and mature viral capsids that are organized by the Gag polyprotein (immature) and by the fully processed CA protein (mature). The Gag polyprotein is composed of three folded polypeptides (MA, CA, and NC) and three smaller peptides (SP1, SP2, and p6) that function together to coordinate membrane binding and Gag-Gag lattice interactions in immature virions. Following budding, HIV maturation is initiated by proteolytic processing of Gag, which induces conformational changes in the CA domain and results in the assembly of the distinctive conical capsid. Retroviral capsids are organized following the principles of fullerene cones, and the hexagonal CA lattice is stabilized by three distinct interfaces. Recently identified inhibitors of viral maturation act by disrupting the final stage of Gag processing, or by inhibiting the formation of a critical intermolecular CA-CA interface in the mature capsid. Following release into a new host cell, the capsid disassembles and host cell factors can potently restrict this stage of retroviral replication. Here, we review the structures of immature and mature HIV virions, focusing on recent studies that have defined the global organization of the immature Gag lattice, identified sites likely to undergo conformational changes during maturation, revealed the molecular structure of the mature capsid lattice, demonstrated that capsid architectures are conserved, identified the first capsid assembly inhibitors, and begun to uncover the remarkable biology of the mature capsid.