  We introduce the concept of `claspers,' which are surfaces in 3-manifolds
with some additional structure on which surgery operations can be performed.
Using claspers we define for each positive integer k an equivalence relation on
links called `C_k-equivalence,' which is generated by surgery operations of a
certain kind called `C_k-moves'. We prove that two knots in the 3-sphere are
C_{k+1}-equivalent if and only if they have equal values of Vassiliev-Goussarov
invariants of type k with values in any abelian groups. This result gives a
characterization in terms of surgery operations of the informations that can be
carried by Vassiliev-Goussarov invariants. In the last section we also describe
outlines of some applications of claspers to other fields in 3-dimensional
topology.
