While students of science and technology are trained to recognise familiar problems with known solutions, they may not be sufficiently prepared to address novel real-world problems. As a step in this direction, we have created and experimented with a new course that is aimed at getting students to think about how to frame and solve unstructured problems. The pedagogical goal is increase students' mathematical awareness and general problem solving skills by employing puzzles, which are educational, engaging, and thought provoking. Over the past year we have run a new course on Puzzle-Based Learning in Australia, United States, and Qatar, with classes from 15 to 380 students. The course is offered in two versions: a full-semester course, and a unit within a general course, such as Introduction to Engineering. In this paper we share our experiences in teaching both courses and discuss our pedagogical objectives, teaching approaches, and coursework. Preliminary observations indicate that the puzzle-based learning approach is assisting students by providing a framework to explore critical thinking, as well as being fun and interesting.