For a given bounded domain with smooth boundary in a smooth Riemannian manifold (M, g), by decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferentialoperator, and by applying Grubb’s method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as t→0+. In particular, we explicitly give the first four coefficients of this asymptotic expansion. These co-efficients provide precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem