Russell, who was born at Trelleck, England, was orphaned at an early age and brought up in the home of his grandfather, the politician Lord John Russell. He was educated privately before attending Cambridge University (1890), from which he graduated (1893) in mathematics. In 1895 he became a fellow and lecturer at Cambridge. His work after 1920 was mainly devoted to the development of his philosophical and political opinions. He became well known for his popularization of many areas of philosophy and also, in works such as The ABC of Atoms (1923) and The ABC of Relativity (1925), of the new trends in scientific thought. For his writings he was awarded the 1950 Nobel Prize for literature. He succeeded his brother to become the 3rd Earl Russell in 1931. Throughout much of his life Russell was an intense advocate of pacifism and during World War I he was imprisoned for expressing these views. Later, in the 1950s and 1960s, he became a central figure in the movements criticizing the use of the atomic bomb, leading demonstrations and mass sit-downs and becoming president of the British Campaign for Nuclear Disarmament in 1958.

At Cambridge Russell became interested in the relatively new discipline of mathematical logic in which he was to be a pioneer. With Guiseppe Peano he was one of the few to recognize the genius of Gottlob Frege and his new system of logic. In 1902 he wrote to Frege, presenting what is now known as Russell's paradox, and asking how Frege's system would deal with it. (Unfortunately, as Frege acknowledged, the system could not accommodate it.) The paradox is one of the paradoxes of set theory and rests on the (then ill-defined) notion of a set. Some sets are members of themselves (the set of all sets is an example because it is itself a set; the set of cats is not an example, as it is not itself a cat). Consider the set of all sets that are not members of themselves: is it a member of itself? If it is, it is not and vice versa. To avoid such paradoxes Russell formulated his logical theory of types. In 1903 he began his collaboration with A. N. Whitehead on their ambitious, if not entirely successful, project of placing mathematics on a sound axiomatic footing by deriving it from logic. This culminated in the publication of Principia Mathematica (1910, 1912, 1913), containing major advances in logic and the philosophy of mathematics.