Measurements from structural brain magnetic resonance imaging (MRI) scans have been increasingly analyzed as intermediate phenotypes to bridge the gap between clinical features and genetic variation. To date, most imaging phenotypes are scalar, such as the volume of a brain region, which can miss subtle or localized morphological variation associated with genetics or relevant to disease. Neuroanatomical shape measurements - multidimensional geometric descriptions of a brain structure - provide an alternate class of phenotypes that remain largely unexplored. In this paper, we extend the concept of heritability to multidimensional traits, and present the first comprehensive analysis of the heritability of neuroanatomical shape measurements across an ensemble of brain structures based on genome-wide single nucleotide polymorphism (SNP) and MRI data from 1,317 unrelated, young (18-35 years) and healthy individuals. Our results demonstrate that neuroanatomical shape can be significantly heritable, above and beyond volume, and thus can serve as a complementary phenotype to study the genetic underpinnings and clinical relevance of brain structure.