We study the almost everywhere convergence of ergodic series of the form $\suma_nf(T^nx)$. This question was first asked by Izumi in 1939 for the casea_n=1/n (called Hilbert transform). Negative answers were given byHalmosfor L^2-convergence (1949) and byErdos-Dowkerfora.e.-convergence (1959). Conditions must be found for the system T, the coefficientsa_nand the function f in order to make the series to converge. A related topic is the convergence of weighted ergodic averages. Joints works with Ch.Cuny, J. Schmeling and W. X. Shen will be presented.