Suppose $X,Y$ are smooth manifolds, $f,g:X\rightarrow Y$ are maps. Then the Coincidence Problem studies the coincidence set $C=\{x:f(x)=g(x)\}$ and $% m=\dim X-\dim Y$ is called the codimension of the problem. For a map $% f:X\rightarrow Z$ and a submanifold $Y$ of $Z,$ the Preimage Problem studies the preimage set $C=\{x:f(x)\in Y\},$ $m=\dim X+\dim Y-\dim Z.$ In case of codimension $0$, the Nielsen number is the lower estimate of the number of points in $C$ changing under homotopies of $f,g,$ and for an arbitrary codimension, of the number of components of $C$. In this talk I will consider an approach to the calculation of other topological characteristics of $C.$ The goal will be to estimate the bordism groups $\Omega _{\ast }(C).$ In comparison to the classical theory the Nielsen equivalence of the points of $C$ is replaced with an equivalence of singular submanifolds of $C$. We consider topologically and algebraically essential classes and define higher order Nielsen numbers.