Methods and structures for monolithically integrating monocrystalline silicon and monocrystalline non-silicon materials and devices are provided. In one structure, a semiconductor structure includes 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 semiconductor structure further includes an insulating layer disposed over the first monocrystalline semiconductor layer in a first region, a monocrystalline silicon layer disposed over the insulating layer in the first region, and 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. The second monocrystalline semiconductor layer has a lattice constant different from the lattice constant of relaxed silicon.