A few eucrites have anomalous oxygen isotopic compositions. To help understand their origin and identify additional samples, we have analyzed the oxygen isotopic compositions of 18 eucrites and four diogenites. Except for five eucrites, these meteorites have ∆ 17 O values that lie within 2σ of their mean value viz., -0.242±0.016‰, consistent with igneous isotopic homogenization of Vesta. The five exceptional eucrites-NWA 1240, Pasamonte (both clast and matrix samples), PCA 91007, A-881394, and Ibitira-have ∆ 17 O values that lie respectively 4σ, 5σ, 5σ, 15σ, and 21σ away from this mean value. NWA 1240 has a δ 18 O value that is 5σ below the mean eucrite value. Four of the five outliers are unbrecciated and unshocked basaltic eucrites, like NWA 011, the first eucrite found to have an anomalous oxygen isotopic composition. The fifth outlier, Pasamonte, is composed almost entirely of unequilibrated basaltic clasts. Published chemical data for the six eucrites with anomalous oxygen isotopic compositions (including NWA 011) exclude contamination by chondritic projectiles as a source of the oxygen anomalies. Only NWA 011 has an anomalous Fe/Mn ratio, but several anomalous eucrites have exceptional Na, Ti, or Cr concentrations. We infer that the six anomalous eucrites are probably derived from five distinct Vesta-like parent bodies (Pasamonte and PCA 91007 could come from one body). These anomalous eucrites, like many unbrecciated eucrites from Vesta, are probably deficient in brecciation and shock effects because they were sequestered in small asteroids (~10 km diameter) during the Late Heavy Bombardment following ejection from Vesta-like bodies. The preservation of Vesta's crust and the lack of deeply buried samples from the hypothesized Vestalike bodies are consistent with the removal of these bodies from the asteroid belt by gravitational perturbations from planets and protoplanets, rather than by collisonal grinding.