Galvanomagnetic measurements have been made at 4.2\ifmmode^\circ\else\textdegree\fi{}K on thin zinc crystal samples, with the thin dimension parallel to the $c$ axis. Different thicknesses in the range 70-400 \ensuremath{\mu} were produced by electropolishing. The magnetic field $H$ was varied between 0 and 14 kG. The longitudinal magnetoresistivity (${\ensuremath{\rho}}_{\ensuremath{\parallel}}$) generally shows no unusual thickness dependence. However, the transverse magnetoresistivity with the field perpendicular to the broad face (${\ensuremath{\rho}}_{\ensuremath{\perp}}$) and the Hall resistivity (${\ensuremath{\rho}}_{21}$) show anomalous thickness dependences. At higher fields, ${\ensuremath{\rho}}_{\ensuremath{\perp}}$ at a given $H$ consistently decreases with diminishing thickness (negative size effect) and obeys an ${H}^{n}$ law, with $n$ nearly 2, at a given thickness. ${\ensuremath{\rho}}_{21}$ shows oscillations, probably magnetomorphic, which are superimposed on a gross Hall resistivity which is approximately proportional to $H$ at higher fields. The gross Hall coefficient $R$ increases rapidly from the thinnest to next thickest sample, then decreases with increasing thickness, becoming slightly negative near 400\ensuremath{\mu}. Sondheimer's size-effect model was extended to a multiparabolic band conductor and evaluated at high fields for both compensated and uncompensated conductors. More general band models are discussed briefly. For a compensated conductor, the parabolic model predicts an inversion of the usual thickness dependence of ${\ensuremath{\rho}}_{\ensuremath{\perp}}$ and a maximum in the magnitude of $R$. The data are in qualititive agreement with these predictions. Physically, these effects are attributed to the interruption by the surface of the carrier trajectories, which prevents the cancellation effects between hole and electron carriers which normally occur at high fields in a compensated conductor. Other measurements in compensated materials are discussed briefly in light of these findings.