Observer’s Mathematics were developed by authors based on denial of infinity idea. In this book authors define the concept of observer, the set of elements where observer works, arithmetic operations depending on observer. There is shown the stochastic appearance in this arithmetic. There are reviewed several classic arithmetical problems from Observer’s Mathematics point of view. Also authors consider elements of algebra, geometry and calculus from Observer’s Mathematics point of view. In particular there are reviewed 10th Hilbert problem and three classic geometries – Euclidean, Gauss-Bolyai-Lobachevsky and Riemannian. Special role plays Observer’s Mathematics black hole concept also considered in this book. And finally authors formulate and prove the Universal Divisibility Criterion working for both arithmetic – classic Mathematics and Observer’s Mathematics.