Abstract: In the talk we discuss the well-known Bernstein-Durrmeyer operators with Jacobi weights on the $d$-dimensional simplices, and their natural quasi-interpolants which were recently introduced by K.~Jetter and J.~Stoeckler. We review some known results and present our recent achievements in such topics as spectral analysis of the operators and the quasi-interpolants, estimates of Jackson-Favard type, direct theorems in terms of appropriate K-functionals as well as complete asymptotic expansions for the operators and the quasi-interpolants, and their derivatives. The talk includes results obtained jointly with K.~Jetter, J.~Stoeckler and U.~Abel.