This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands and deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. The paper provides noncommutative generalizations of classical bounds associated with Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities offer the same diversity of application, ease of use, and strength of conclusion as their scalar counterparts. The main technical contribution is a deep theorem by Lieb on convex trace functions, which is used to obtain a more satisfactory framework for bounding the matrix moment-generating function. This approach leads to a collection of probability inequalities that are essentially sharp in various situations, including matrix Gaussian and Rademacher series, and sums of rectangular matrices. The paper also discusses related work, including the matrix Laplace transform method and noncommutative moment inequalities, and provides a roadmap for the rest of the article.This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands and deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. The paper provides noncommutative generalizations of classical bounds associated with Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities offer the same diversity of application, ease of use, and strength of conclusion as their scalar counterparts. The main technical contribution is a deep theorem by Lieb on convex trace functions, which is used to obtain a more satisfactory framework for bounding the matrix moment-generating function. This approach leads to a collection of probability inequalities that are essentially sharp in various situations, including matrix Gaussian and Rademacher series, and sums of rectangular matrices. The paper also discusses related work, including the matrix Laplace transform method and noncommutative moment inequalities, and provides a roadmap for the rest of the article.