This textbook introduces the central debates in the philosophy of mathematics. It is aimed at upper-level undergraduates with a limited proficiency in mathematics, but who have completed a first-year formal logic course. The book examines the standard theories of mathematics: platonism and realism, logicism, formalism, constructivism and structuralism, as well as some less standard theories, such as psychologism, fictionalism, and Meinongian philosophy of mathematics. In each case, the author explains what characterises the position, where the divisions between them lie, and what are some of the arguments in favour and against the positions. The book also explores particular questions that occupy present-day philosophers and mathematicians, such as the relationship between good reasoning and mathematics, the problem of infinity and whether we are more certain of mathematics than we are of everyday sense experience, or science. The book aims to strike a balance between conceptual accessibility and correct representation of the issues in the philosophy of mathematics in order for the student to acquire the courage, resources and curiosity to challenge existing positions.