Nonpremixed flames subject to steady but spatially varying straining flows were studied to examine one aspect of nonpremixed flames in strongly turbulent flows or near quenching conditions (e.g., near a burner rim), where strain-rate gradients are present and local strain rates may be high enough to cause local flame-front extinguishment. The spatially varying straining flows were created using an opposed slot-jet burner with slightly nonparallel jet exits. The most significant observation was that steady flame "edges" could be created where the flame would exist in the low-strain region but would be extinguished in the high-strain region. The local strain at the location of the stationary flame edge was almost always lower than the strain required to extinguish flames in the same mixture subject to a spatially uniform strain. The strain rate at the edge-flame location was independent of the strain-rate gradient and gradual transitions from edge-flame behavior to uniformly strained flame behavior were not observed, indicating that conventional nonpremixed flames and edge flames are quite distinct structures yet each has well-defined properties. At the flame edge, interferometer images indicate a region of locally intense burning and an abrupt transition to nonburning conditions away from the edge. These observations were found to be qualitatively consistent with recent theoretical models of flame edges. These results indicate that "laminar flamelet" models of nonpremixed turbulent combustion may require reevaluation at conditions approaching those where local flame quenching occurs. It is proposed that these models could be improved by adding edge-flame libraries to existing laminar flamelet libraries.