Chaos Detection and Predictability Estimating Lyapunov exponents from time series Theory and applications of the fast Lyapunov Indicator (FLI) method Theory and applications of the Orthogonal Fast Lyapunov Indicator (OFLI and OFLI2) methods Theory and applications of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) method The Smaller (SALI) and the Generalized (GALI) Alignment Indices: Efficient Methods of Chaos Detection The Relative Lyapunov Indicators: Theory and Application to Dynamical Astronomy The 0 1 Test for Chaos: A review. Distinguishing chaoticity from regularity in deterministic dynamical systems and specifying the subspace of the phase space in which instabilities are expected to occur is of utmost importance in as disparate areas as astronomy particle physics and climate dynamics. To address these issues there exists a plethora of methods for chaos detection and predictability. The most commonly employed technique for investigating chaotic dynamics i.e. the computation of Lyapunov exponents however may suffer a number of problems and drawbacks for example when applied to noisy experimental data. In the last two decades several novel methods have been developed for the fast and reliable determination of the regular or chaotic nature of orbits aimed at overcoming the shortcomings of more traditional techniques. This set of lecture notes and tutorial reviews serves as an introduction to and overview of modern chaos detection and predictability techniques for graduate students and non specialists. The book covers theoretical and computational aspects of traditional methods to calculate Lyapunov exponents as well as of modern techniques like the Fast (FLI) the Orthogonal (OFLI) and the Relative (RLI) Lyapunov Indicators the Mean Exponential Growth factor of Nearby Orbits (MEGNO) the Smaller (SALI) and the Generalized (GALI) Alignment Index and the ‘0 1’ test for chaos.