On the strong law of large numbers for sequences of dependent random variables with finite second moments

New sufficient conditions of a.s. convergence of the series $\sum_{n=1}^\infty X_n$ and new sufficient conditions for the applicability of the strong law of large numbers are established for a sequence of dependent random variables $\{X_n\}_{n=1}^\infty$ with finite second moments. These results are generalizations of the well known theorems on a.s. convergence of the series of orthogonal random variables and on the strong law of large numbers for orthogonal random variables (Men'shov–Rademacher and Petrov's theorems). It is shown that some of the results obtained are optimal.