Methods and structures for monolithically integrating monocrystalline silicon and monocrystalline non-silicon materials and devices are provided. In one structure, a monolithically integrated semiconductor device structure comprises a silicon substrate and a first monocrystalline semiconductor layer disposed over the silicon substrate, wherein the first monocrystalline semiconductor layer has a lattice constant different from a lattice constant of relaxed silicon. The structure further includes an insulating layer disposed over the first monocrystalline semiconductor layer in a first region and a monocrystalline silicon layer disposed over the insulating layer in the first region. The structure includes at least one silicon-based photodetector comprising an active region including at least a portion of the monocrystalline silicon layer. The structure also includes a second monocrystalline semiconductor layer disposed over at least a portion of the first monocrystalline semiconductor layer in a second region and absent from the first region, wherein the second monocrystalline semiconductor layer has a lattice constant different from the lattice constant of relaxed silicon. The structure includes at least one non-silicon photodetector comprising an active region including at least a portion of the second monocrystalline semiconductor layer.