The principal object of this work is to give an extensive coherent treatment of ergodic theorems for weighted averages and ergodic theorems along subsequences. In other words, the authors study the question for which sequences
of complex numbers and for which operators in
the averages (1/n)
converge a.e. for all
. A related question is for which strictly increasing sequences
of integers the averages (1/n)
converge a.e. The paper also contains a number of new results, and suggests further avenues of research.