We study a class of micromanipulation experiments, exemplified by the pulling apart of the two strands of double-stranded DNA. When the pulling force is increased to a critical value, an "unzipping" transition occurs. For random DNA sequences with short-ranged correlations, we obtain exact results for the number of monomers liberated and the specific heat, including the critical behavior at the transition. Related systems include a random heteropolymer pulled away from an adsorbing surface and a vortex line in a type II superconductor tilted away from a fragmented columnar defect.