Asymptotic estimates of certain Meijer G-functions are derived using contour integration techniques. Usually one or several of the parameters are large, while the arguments and some other parameters serve as uniformity parameters. For special values of the parameters a result is otained for the Jacobi polynomials, which agrees with the classical expansion of Darboux, but it is valid in a wider domain than earlier results are. The paper gives a very detailed analysis of the problem and an extensive demonstration of the machinery of complex variable techniques for contour integrals over Riemann sheets of multi-valued functions.