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 a corollary. The proof techniques also yield information about matrix-valued martingales. The paper provides noncommutative generalizations of classical bounds associated with Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. These matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion as their scalar counterparts. The paper introduces a new framework for bounding the matrix moment-generating function (mgf). The key innovation is a theorem by Lieb that allows the derivation of a wide range of probability inequalities. The paper presents several main results, including matrix versions of the Chernoff, Hoeffding, and Bernstein inequalities. These results provide tail bounds for the maximum eigenvalue of sums of random matrices and for the maximum singular value of sums of random rectangular matrices. The paper also discusses matrix martingales and provides a detailed analysis of the techniques used to derive these inequalities. The results are shown to be structurally optimal and often provide better bounds than previous methods. The paper concludes with a discussion of related work and a roadmap of the rest of the paper.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 a corollary. The proof techniques also yield information about matrix-valued martingales. The paper provides noncommutative generalizations of classical bounds associated with Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. These matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion as their scalar counterparts. The paper introduces a new framework for bounding the matrix moment-generating function (mgf). The key innovation is a theorem by Lieb that allows the derivation of a wide range of probability inequalities. The paper presents several main results, including matrix versions of the Chernoff, Hoeffding, and Bernstein inequalities. These results provide tail bounds for the maximum eigenvalue of sums of random matrices and for the maximum singular value of sums of random rectangular matrices. The paper also discusses matrix martingales and provides a detailed analysis of the techniques used to derive these inequalities. The results are shown to be structurally optimal and often provide better bounds than previous methods. The paper concludes with a discussion of related work and a roadmap of the rest of the paper.