This paper discusses the properties of log-concave probability distributions and their applications. It presents a series of theorems relating log-concavity and log-convexity of probability density functions, distribution functions, reliability functions, and their integrals. The authors list many commonly-used probability distributions and report whether their density functions and integrals are log-concave or log-convex. They also discuss various applications of log-concavity in economics, political science, biology, and industrial engineering. A function is log-concave if the natural logarithm of the function is concave, and log-convex if it is convex. The paper examines the invariance of these properties under integration, truncations, and other transformations. It relates the properties of density functions to those of reliability functions, failure rates, and the monotonicity of the "mean-residual-lifetime function." The authors define the "mean-advantage-over-inferior function" for truncated distributions and relate its monotonicity to log-concavity or log-convexity of the probability density function and its integral. The paper examines many commonly-used probability distributions and records the log-concavity or log-convexity of their density functions and integrals. It also discusses various applications of log-concavity that have appeared in the literature. Most of the results in this paper have appeared in the literature of statistics, economics, and industrial engineering. The purpose of this paper is to offer a unified exposition of related results on the log-concavity and log-convexity of univariate probability distributions and to sample some applications of this theory. An earlier draft of this paper was available on the web since 1989. The current version streamlines the exposition and proofs and makes note of several related papers that have appeared since 1989. Theorem 1 states that if a probability density function is continuously differentiable and log-concave on an interval (a, b), then its corresponding cumulative distribution function is also log-concave on that interval. This result was proved by Prékopa, and the paper provides a simple calculus proof.This paper discusses the properties of log-concave probability distributions and their applications. It presents a series of theorems relating log-concavity and log-convexity of probability density functions, distribution functions, reliability functions, and their integrals. The authors list many commonly-used probability distributions and report whether their density functions and integrals are log-concave or log-convex. They also discuss various applications of log-concavity in economics, political science, biology, and industrial engineering. A function is log-concave if the natural logarithm of the function is concave, and log-convex if it is convex. The paper examines the invariance of these properties under integration, truncations, and other transformations. It relates the properties of density functions to those of reliability functions, failure rates, and the monotonicity of the "mean-residual-lifetime function." The authors define the "mean-advantage-over-inferior function" for truncated distributions and relate its monotonicity to log-concavity or log-convexity of the probability density function and its integral. The paper examines many commonly-used probability distributions and records the log-concavity or log-convexity of their density functions and integrals. It also discusses various applications of log-concavity that have appeared in the literature. Most of the results in this paper have appeared in the literature of statistics, economics, and industrial engineering. The purpose of this paper is to offer a unified exposition of related results on the log-concavity and log-convexity of univariate probability distributions and to sample some applications of this theory. An earlier draft of this paper was available on the web since 1989. The current version streamlines the exposition and proofs and makes note of several related papers that have appeared since 1989. Theorem 1 states that if a probability density function is continuously differentiable and log-concave on an interval (a, b), then its corresponding cumulative distribution function is also log-concave on that interval. This result was proved by Prékopa, and the paper provides a simple calculus proof.