Working physicists, and especially astrophysicists, value a good `back-of-the-envelope' calculation, meaning a short, elegant computation or argument that starts from general principles and leads to an interesting result. This book guides students on how to understand astrophysics using general principles and concise calculations - endeavouring to be elegant where possible and using short computer programs where necessary. The material proceeds in approximate historical order. The book begins with the Enlightenment-era insight that the orbits of the planets is easy, but the orbit of the Moon is a real headache, and continues to deterministic chaos. This is followed by a chapter on spacetime and black holes. Four chapters reveal how microphysics, especially quantum mechanics, allow us to understand how stars work. The last two chapters are about cosmology, bringing us to 21st-century developments on the microwave background and gravitational waves.