Structures including an edge coupler and methods of fabricating a structure including an edge coupler. The edge coupler includes a waveguide core, and a shaped layer is positioned over a portion of the waveguide core. The waveguide core is comprised of a first material, and the shaped layer is comprised of a second material different in composition from the first material. The first material may be, for example, single-crystal silicon, and the second material may be, for example, silicon nitride.