Spectra of asteroid 4 Vesta and 21 small (estimated diameters less than 10 km) asteroids with Vesta-like spectral properties (Vestoids) were measured at visible and near-infrared wavelengths (similar to0.44 to similar to1.65 mum). All of the measured small asteroids (except for 2579 Spartacus) have reflectance spectra consistent with surface compositions similar to eucrites and howardites and consistent with all being derived from Vesta. None of the observed asteroids have spectra similar to diogenites. We find no spectral distinction between the 15 objects tabulated as members of the Vesta dynamical family and 6 of the 7 sampled "non-family" members that reside just outside the semi-major axis (a), eccentricity (e), and inclination (i) region of the family. The spectral consistency and close orbital (a-e-i) match of these "non-family" objects to Vesta and the Vesta family imply that the true bounds of the family extend beyond the subjective cut-off for membership. Asteroid 2579 Spartacus has a spectrum consistent with a mixture of eucritic material and olivine. Spartacus could contain olivine-rich material from Vesta's mantle or may be unrelated to Vesta altogether. Laboratory measurements of the spectra of eucrites show that samples having nearly identical compositions can display a wide range of spectral slopes. Finer particle sizes lead to an increase in the slope, which is usually referred to as reddening. This range of spectral variation for the best-known meteoritic analogs to the Vestoids, regardless of whether they are actually related to each other, suggests that the extremely red spectral slopes for some Vestoids can be explained by very fine-grained eucritic material on their surfaces.