Stieltjes' constants $\gamma_n$ are the coefficients in the Laurent series for the zeta function $\zeta(s)$ at the pole $s=1$. We present new sequences of approximations for Stieltjes' constants obtained by generalizing the ``remainder Pad\'e approximants'' method introduced by the first named author in 1996. Here, we replace Pad\'e approximants (of which poles are connected to zeros of orthogonal polynomials) by Pad\'etype approximants introduced by Brezinski of which poles are chosen {\em a priori}. The particular case $\gamma_1$ is also treated separately using ordinary Pad\'e approximants. The last section of the paper deals with approximations of the Riemann zeta function in the complex plane.