Enumerative geometry is traditionally considered as a
subject of intersection theory. Modern intersection theory with its
methods of singular schemes, blowing-ups, residue intersections, and
Hilbert schemes is rather technical and a rigorous proof of very
simple geometrically evident results require often many pages of
complicated formulas and commutative diagrams.

For example, the number of planes in $\C P^3$ touching a fixed degree
$d$ smooth hypersurface at three points was computed by Salmon in the
middle of XIX-th century but the proof that would satisfy
modern algebraists has appeared comparatively recently. On the other
hand, the answer to the similar question about hypersurfaces in $\C P^n$,
for $n>3$ seems to be unknown.

In this note we present an alternative approach to enumerative
geometry whose motivation belongs to singularity theory and topology,
more precisely, to the theory of cobordisms and cohomological
operations. We announce a universal formula that expresses the
classes dual to various multisingularity loci via so called residue
polynomials associated with the multisingularities. This formula
refines the known multiple point formulas due to
S.L.~Kleiman~\cite{Kl1,Kl2,Kl3} and S.~Katz~\cite{Katz} to the case
of arbitrary proper maps and generalizes it to the case of arbitrary
multisingularity types.

The formula reduces the problem of finding classes of
multisingularity loci to the problem of finding the coefficients of
the residue polynomial. To find these coefficients we use a method of
using symmetries suggested by R.~Rim\'anyi for a close
problem~\cite{R1,R2}. Note that these computations are very far from
the concrete geometrical problems to which the formulas are applied.
This is the reason, perhaps, why our main formula have not been
observed before.

We present also a version of this formula for the classes dual to the
loci of isolated hypersurface multisingularities. Particular cases of
this formula cover many classical enumerative problems: enumerating
singular curves on surfaces, enumerating singular hyperplane sections
of a fixed projective variety, enumerating projective subspaces of a
given tangency type with a fixed hypersurface and many others. Our
computations agree with the classical formulas of Pl\"ucker and
Salmon as well as with more recent results of P.~Le Barz~\cite{LB},
S.J.~Colley~\cite{Co1,Co2}, V.A.~Vainsencher~\cite{Vains},
L.~G\"ottsche~\cite{G}, L.~Caporaso and J.~Harris~\cite{CH},
S.L.~Kleiman and R.~Piene~\cite{KP1,KP2} and many other authors
contributed to the solution of these problems. In particular, our
formula confirms and refines the G\"ottsche's conjecture on the
number of plane nodal curves and all its variations due to Kleiman
and Piene. 

We recognize that the present paper contains no proof of the main
formula. Therefore until the complete proof is published it should
be considered as a conjecture. But after numerous topological and
enumerative confirmations there is no doubt that it is valid also
for Chow rings and over any algebraically closed ground field.