THE critical current density across individual grain boundaries in thin films of the high-T(c) superconductor YBa2Cu3O7-delta (YBCO) has been found 1-4 to be inversely proportional to lattice misorientation for tilts up to approximately 10-degrees. Reports of impurity segregation 5,6 at grain boundaries, and variations in the chemical stoichiometry 7,8, have led to the view that deviations from the ideal composition are responsible for the depressed superconducting order parameter at the boundary. Here we present images of YBCO grain boundaries obtained by a scanning transmission electron microscope in Z-contrast mode 9,10, which show that chemical segregation does not necessarily occur at these boundaries. A simple model of the strain associated with the grain-boundary dislocations provides a reasonable physical explanation of the suppressed superconductivity. The surprisingly large effect of strain implied by our model has implications beyond critical currents, for the physics and applications of any thin-film YBCO structures involving strained epitaxial layers.