Alain Badiou is one of the most inventive and compelling philosophers working in France today-a thinker who, in these days of cynical resignation and academic specialization, is exceptional in every sense. Guided by disciplines ranging from mathematics to psychoanalysis, inspired as much by Plato and Cantor as by Mao and Mallarmé, Badiou's work renews, in the most varied and spectacular terms, a decidedly ancient understanding of philosophy-philosophy as a practice conditioned by truths, understood as militant processes of emancipation or transformation. This book is the first comprehensive introduction to Badiou's thought to appear in any language. Assuming no prior knowledge of his work, it provides a thorough and searching overview of all the main components of his philosophy, from its decisive political orientation through its startling equation of ontology with mathematics to its resolute engagement with its principal competition (from Wittgenstein, Heidegger, and Deleuze, among others). The book draws on all of Badiou's published work and a wide sampling of his unpublished work in progress, along with six years of correspondence with the author.  Peter Hallward pays careful attention to the aspect of Badiou's work most liable to intimidate readers in continental philosophy and critical theory: its crucial reliance on certain key developments in modern mathematics. Eschewing unnecessary technicalities, Hallward provides a highly readable discussion of each of the basic features of Badiou's ontology, as well as his more recent account of appearance and "being-there." Without evading the difficulties, Peter Hallward demonstrates in detail and in depth why Badiou's ongoing philosophical project should be recognized as the most resourceful and inspiring of his generation.