Grayling offers a survey of Russell’s life, his contributions to logic and analytic philosophy, his work on mind and science, and his role as a public intellectual engaged with politics, war, education, and morality. He ends with a brief overview and assessment of Russell’s continuing influence on philosophy.
This is a helpful introduction. Grayling manages to balance his presentation, showing Russell not only as a brilliant logician (I first encountered Russell in his [and Whitehead’s] Principia Mathematica), but also as a man shaped by loss, conviction, controversy, and extensive public discourse (Grayling hardly mentions Why I am Not a Christian, but that is another book that introduced me to Russell). Because Grayling focuses not only on Russell’s philosophical work but also on his social activism, the book is more engaging than I expected in an academic summary.
That said, at only 150 pages (it is a "very short introduction" after all) the book trades depth for breadth. Russell’s philosophical positions are briefly sketched and some of the tensions in his thought, though referenced, are truncated by the need for brevity. That is not meant as a criticism, it’s just the natural limitation of the series. If you want to read and understand his sustained philosophical arguments, you will still need to read Russell himself. Thankfully, this book is a valuable gateway to his thought, because, as always, the bibliography following each chapter is an invaluable starting point for further study.