A theory of the scaling behavior of the thermodynamic, transport and 
dynamical properties of a three-dimensional metal at an 
antiferromagnetic (AFM) critical point is presented. It is show how
the critical spin fluctuations at the AFM wavevector q = Q induce energy 
fluctuations at small q, giving rise to a diverging quasiparticle 
effective mass over the whole Fermi surface. The coupling
of the fermionic and bosonic degrees of freedom leads to a 
self-consistent relation for the effective mass, which has a strong 
coupling solution in addition to the well-known weak-coupling 
spin-density wave solution. We thereby use the recently introduced 
concept of critical quasiparticles, employing a scale dependent 
effective mass ratio m*/m and quasiparticle weight factor Z. We 
specifically assume a scale dependent vertex correction boosting the 
coupling of fermions and spin fluctuations. The ensuing spin fluctuation 
spectrum obeys E/T-scaling. Our results are in good agreement with
experimental data on the heavy fermion compounds YbRh2Si2 and CeCu6-xAux 
assuming 3D and 2D spin fluctuations, respectively.