The proposal, revived by Tremaine and Zytkow (1985), that accretion of comets by neutron stars may be the origin of gamma-ray bursts is considered. This mechanism has difficulty accounting for the observed gamma-ray spectrum and optical counterparts of the bursts. The survival of comets near supernovae is investigated. Ablation rates and the thermal structure of an ablating surface layer are calculated. In some circumstances, mechanical disruption will erode a comet more rapidly than evaporation. The accretion of comets by neutron stars may produce a class of X-ray burst sources with novel properties.