This open access book is intended to give a bird's-eye view of ellipses and planetary orbits. The only background required is secondary-school Euclidean geometry, analytic geometry, and trigonometry. That doesn't mean that the theorems and proofs are easy; to the contrary, many are very challenging.Although Isaac Newton invented the calculus and used it to study motion, from the time of the Greeks, proof meant proof by geometry. The book contains Newton's detailed geometric proof of the inverse-square law of orbits, based on Conic Sections Treated Geometrically, a widely used textbook from the nineteenth century written by William H. Besant. An important feature of the book is the numerous diagrams that are much more detailed than those appearing in the textbooks from the nineteenth century.Turning to planetary orbits, the book presents Kepler's equation for computing the position, speed and direction of a planet in its orbit, followed by the computation of Lagrange points, which are points in the solar system where a spacecraft can be placed so that the period of its orbit is the same as the Earth's.The history of mathematics has (or should have) an important place in mathematics education. Euclid is well-known but mathematicians were equally familiar with Conics by Apollonius of Perga. Some of his results are given in modern notation, although the presentation is faithful to his style. In addition, Kepler's own geometric proof of his First Law is given.The final chapter presents challenging theorems on ellipses: the Steiner inellipse, Marden's Theorem, the theorems of Pascal and Brianchon, and Newton's Ellipse Theorem.