Multijunction solar cells can be fabricated by bonding together component cells that are grown separately. Because the component cells are each grown lattice-matched to suitable substrates, this technique allows alloys of different lattice constants to be combined without the structural defects introduced when using metamorphic buffers. Here we present results on the fabrication and performance of four-junction mechanical stacks composed of GaInP/GaAs and GaInAsP/GaInAs tandems, grown on GaAs and InP substrates, respectively. The two tandems were bonded together with a low-index, transparent epoxy that acts as an omni-directional reflector to the GaAs bandedge luminescence, while simultaneously transmitting nearly all of the sub-bandgap light. As determined by electroluminescence measurements and optical modeling, the GaAs subcell demonstrates a higher internal radiative limit and thus higher subcell voltage, compared with GaAs subcells without enhanced internal optics; all four subcells exhibit excellent material quality. The device was fabricated with four contact terminals so that each tandem can be operated at its maximum power point, which raises the cumulative efficiency and decreases spectral sensitivity. Efficiencies exceeding 38% at one-sun have been demonstrated. Eliminating the series resistance is the key challenge for the concentrator cells. We will discuss the performance of one-sun and concentrator versions of the device, and compare the results to recently fabricated monolithic four-junction cells.