Mathematical logic has grown from an exotic branch of mathematics into an indispensable tool in computer science as well as other parts of mathematics. This concise book presents the subject of mathematical logic in a lively and approachable fashion although logic can be a formidably abstruse topic, even for mathematicians. This second edition of What is Mathematical Logic?, originally published 50 years ago, deals with important ideas in modern mathematical logic, without the detailed mathematical work required of those with a professional interest in logic. The ideas are set forth simply and clearly in a pleasant style and, despite the book's relative brevity, all the basic material is covered in these pages. Three new chapters have been added, covering automatic theorem proving, logic beyond traditional first order logic, and other logics including intuitionistic, free, and modal logics. Students of computer science and mathematical logic will find it a stimulating introduction and valuable supplement for courses, including current further reading suggestions in this lively area at the intersection of mathematics, philosophy, and computer science.