For weighted sums, general strong laws of large numbers of the formalmost certainly are established where the random variablesare stochastically dominated by a random variableare suitable conditional expectations or are all 0. The hypotheses involve both the behavior of the tail of the distribution of ∣Y∣ and the growth behavior of the constants bn/an. In general, no assumptions are made regarding the joint distributions of. As special cases, both old and new results are obtained