Given a generic PL map or a generic smooth fold map \(f\colon N^n\to M^m\) , where \(m\ge n\) and \(2(m+k)\ge 3(n+1)\) , we prove that f lifts to a PL or smooth embedding \(N\to M\times \mathbb {R}^k\) if and only if its double point locus \(\{(x,y)\in N\times N\mid f(x)=f(y),\,x\ne y\}\) admits an equivariant map to \(S^{k-1}\) . As a corollary we answer a 1990 question of P. Petersen and obtain some other applications. We also discuss several criteria for lifting a non-degenerate PL map or a \(C^0\) -stable smooth map \(f\colon N^n\to M^m\) , where \(m\ge n\) , to an embedding in \(M\times \mathbb {R}\) , elaborating on V. Poénaru’s observations. In particular, the existence of such a lift is determined by the equivariant homotopy type of the diagram consisting of the three projections from the triple point locus \(\{(x,y,z)\in N\times N\times N\mid f(x)=f(y)=f(z),\,x\ne y\ne z\ne x\}\) to the double point locus. The three Appendices, which can be read independently of the rest of this chapter, are devoted to stable and generic maps. Appendix B introduces an elementary theory of stable PL maps. Appendix C extends the 2-multi-0-jet transversality theorem over the usual compactification of \(M\times M\setminus \Delta _M\) .