Simple and mechanical systems are the fouundation of Science: a particle sliding on a two-dimensional surface in mathematics, the solar system in astronomy, the hydrogen and the helium atoms in physics, and small molecules in chemistry. Although elementary and deterministic, their motions look almost random over long time intervals, and cannot be explained by the traditional approaches. This book describes the apparent chaos in these systems for an audience at the level of beginning graduate students, developing the relevant ideas of the last two decades with the help of geometric intutition rather than algebraic manipulation. The main focus is on seeking the connection between classical and quantum mechanics: classical chaos is rough and fractal, whereas quantum chaos is smooth and elusive; and yet the former should be the limit of the latter as Planck's quantum becomes small. The historical and cultural background is mentioned, and the text discusses realistic examples in some detail thereby providing readers a new perspective and preparing them to tackle new problems in this rich field.