We consider the Robin eigenvalue problem ∆u + λu = 0 in Ω, ∂u/∂ν + αu = 0 on ∂Ω where Ω ⊂ R n , n > 2 is a bounded domain and α is a real parameter. We investigate the behavior of the eigenvalues λk(α) of this problem as functions of the parameter α. We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative λ ′ 1 (α). Assuming that the boundary ∂Ω is of class C 2 we obtain estimates to the difference λ D k −λk(α) between the k-th eigenvalue of the Laplace operator with Dirichlet boundary condition in Ω and the corresponding Robin eigenvalue for positive values of α for every k = 1, 2, . . ..