An electronic device may be provided with a display. The display may be formed from liquid crystal display pixels, organic light-emitting diode pixels, or other pixels. The display may have an active area that is bordered along at least one edge by an inactive area. The active area contains pixels and displays images. The inactive area does not contain any pixels and does not display images. The inactive area may have a layer of black ink or other masking material to block internal components from view. The active area may have an opening that contains an isolated portion of the inactive area or may contain a recess into which a portion of the inactive area protrudes. An electrical component such as a speaker, camera, light-emitting diode, light sensor, or other electrical device may be mounted in the inactive area in the recess or opening of the active area.