This book contains around of a score of essays that examine or use Popperís fallibilist philosophy of science from various points of view. The opening sextet includes a new solution of the problem of induction, the status of corroboration, and the place of truth-values and logic(s). Then follow a trio of studies of relationships with some other philosophies, especially Bertrand Russellís. Next come five essays on the history and historiography of science and mathematics, and of mathematics education. They are followed by a quintet of speculations relating to aspects of psychology, especially psychical research, coincidences and the formation of languages in humans. Finally come two new items: one on the philosophy of mathematics, the other a public acknowledgement of the influence of Popper in August 2009 when the International Commission on the History of Mathematics awarded Grattan-Guinness the Kenneth O. May Medal and Prize in the History of Mathematics. I. Grattan-Guinness is Emeritus Professor of the History of Mathematics and Logic at Middlesex University, England, and also a Visiting Research Associate in the Centre for Philosophy of Natural and Social Science at the London School of Economics.