The dynamics of the deformation of a drop in axisymmetric compressional viscous flow is addressed through analytical and numerical analyses for a variety of capillary numbers,
$\mathit{Ca}$, and viscosity ratios,
$\lambda $. For low
$Ca$, the drop is approximated by an oblate spheroid, and an analytical solution is obtained in terms of spheroidal harmonics; whereas, for the case of equal viscosities (
$\lambda = 1$), the velocity field within and outside a drop of a given shape admits an integral representation, and steady shapes are found in the form of Chebyshev series. For arbitrary
$Ca$ and
$\lambda $, exact steady shapes are evaluated numerically via an integral equation. The critical
$\mathit{Ca}$, below which a steady drop shape exists, is established for various
$\lambda $. Remarkably, in contrast to the extensional flow case, critical steady shapes, being flat discs with rounded rims, have similar degrees of deformation (
$D\sim 0. 75$) for all
$\lambda $ studied. It is also shown that for almost the entire range of
$\mathit{Ca}$ and
$\lambda $, the steady shapes have accurate two-parameter approximations. The validity and implications of spheroidal and two-parameter shape approximations are examined in comparison to the exact steady shapes.