In 1983 E. M. Stein proved that the vector of classical Riesz transforms has $L^p$ bounds on $\mathbb R^d$ which are independent of the dimension. I will discuss an analogous result for the vector of maximal Riesz transforms. I will also mention generalizations to higher order Riesz transforms. A main principle of our approach is a comparison between a maximal Riesz transform and the corresponding Riesz transform. The talk is based on joint work with Maciej Kucharski and Jacek Zienkiewicz (Wrocław).