We investigate asymptotics of probabilities of moderate deviations and their logarithms for an array of row-wise independent random variables with finite variations and finite one-sided moments of order p > 2. The range of the zone of normal convergence is calculated in terms of Lyapunov ratios constructed from the positive parts of the random variables. Bounds for probabilities of moderate deviations are also derived in the case where the normal convergence fails. Bibliography: 16 titles.