Order stars is a recently developed technique to analyze and explain the behaviour of numerical methods. The main idea is to explore different features of numerical algorithms as properties of analytical functions in various portions of the complex plane. Thus, for example, the order of some numerical methods for ordinary differential equations can be translated to the language of approximation theory - specifically, to the question of how well a given rational function R approximates the exponential. Likewise, stability properties of the underlying method can be expressed as some other features of the function R. In this formulation, order stars establish the relationship between order and stability, helping in the search for better and more efficient computational algorithms. The book describes in detail the application of order stars to ordinary, parabolic and hyperbolic differential equations, as well as to approximation theory. This book should be of interest to theoretically inclined numerical analysts (mainly in academic institutions, but also in industry) and mathematical analysts (mainly approximation theorists and specialists in analytic functions).