V.A.Zagrebnov Trotter-Kato Product Formula: Some Recent Results (175K, Postscript) ABSTRACT. Recently new results about the operator-norm and the trace-norm convergences of the Trotter-Kato product formulae with ultimate optimal error-bound estimates have been established. They concern the self-adjoint strongly continuous and Gibbs semigroups on Hilbert spaces. Explicit dependence of the rate of convergence on the generators domains is found.