  In terms of category theory, the Gromov homotopy principle for a set valued
functor $F$ asserts that the functor $F$ can be induced from a homotopy
functor. Similarly, we say that the bordism principle for an abelian group
valued functor $F$ holds if the functor $F$ can be induced from a (co)homology
functor. We examine the bordism principle in the case of functors given by
(co)bordism groups of maps with prescribed singularities. Our main result
implies that if a family $R$ of prescribed singularity types satisfies certain
mild conditions, then there exists an infinite loop space $B(R)$ such that for
each smooth manifold $N$ the cobordism group of maps into $N$ with only
$R$-singularities is isomorphic to the group of homotopy classes of maps $[N,
B(R)]$.
