  We establish a one-to-one correspondence between virialized haloes and their
seeds, namely peaks with a given density contrast at appropriate
Gaussian-filtering radii, in the initial Gaussian random density field. This
fixes a rigorous formalism for the analytic derivation of halo properties from
the linear power spectrum of density perturbations in any hierarchical
cosmology. The typical spherically averaged density profile and mass function
of haloes so obtained match those found in numerical simulations.
