  This paper is devoted to the study of the log-convexity of combinatorial
sequences. We show that the log-convexity is preserved under componentwise sum,
under binomial convolution, and by the linear transformations given by the
matrices of binomial coefficients and Stirling numbers of two kinds. We develop
techniques for dealing with the log-convexity of sequences satisfying a
three-term recurrence. We also introduce the concept of $q$-log-convexity and
establish the connection with linear transformations preserving the
log-convexity. As applications of our results, we prove the log-convexity and
$q$-log-convexity of many famous combinatorial sequences of numbers and
polynomials.
