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Recent advances in bonding and thin-layer splitting technologies enables a new approach to integrate dissimilar semiconductor materials and microsystems. The integration process incorporates the bonding of two substrates and the use of delamination to achieve wafer-scale or die-scale layer transfer. This 'paste-and-cut' integration method is an appealing alternative to direct deposition approaches because each material or microsystem layer for a given device function can be grown or fabricated on an ideally suited substrate and then combined with a dissimilar receptor substrate. In this paper, we review available technologies for pasting (direct bonding, anodic bonding, metal bonding, and polymeric bonding) and for cutting (ion cut, chemical cut, mechanical cut, and laser liftoff). Advanced semiconductor substrates, MEMS, and photonics-on-Si are used as examples to illustrate the versatility of this approach.