  In this paper, by explicitly calculating the principal symbols of
pseudodifferential operators and by applying H\"omander's spectral function
theorem, we obtain the Weyl-type asymptotic formulas with sharp remainder
estimates for the counting functions of the two classes of biharmonic Steklov
eigenvalues $\lambda_k$ and $\mu_k$ in a smooth bounded domain of a Riemannian
manifold. This solves a longstanding challenging problem.
