The dynamics of viscous drops in linear creeping flows are investigated near the critical flow strength at which stationary drop shapes cease to exist. According to our theory the near-critical behavior of drops is dominated by a single slow mode evolving on a time scale that diverges at the critical point with exponent 1/2. The theory is based on the assumption that the system undergoes a saddle-node bifurcation. The predictions have been verified by numerical simulations for drops in axisymmetric straining flow and in two-dimensional flows with less vorticity than in shear flow. Application of our theory to the accurate determination of critical parameters is discussed.