A hypersubstitution is a mapping from the set of operation symbols into terms which assign to an \(n\)-ary operation symbol an \(n\)-ary term and which can be uniquely extended to mappings defined on the set of all terms. It is shown that kernels of hypersubstitutions are fully invariant congruences on an absolutely free algebra of the corresponding type. In the case when a type contains just one \(n\)-ary operation symbol, the authors get a description of all these congruences. The result is also generalized to arbitrary types, and it is used for solving the hyperunification problem.