This monograph deals with random forests and aims to show that, despite the outward simplicity of the forest graph design, the problems emerging in relation to the phenomenon are challenging, and their solution often requires subtle mathematical methods. The book focuses on forests formed by simply generated trees. While the notion of the simply generated family of trees covers many known tree classes, this approach to the study of random forests gives a chance to obtain results for various forest types in a uniform way. The main objective of the book is to provide the full description of the limit behaviour of the random forest's most important characteristics: maximum tree size and number of trees of a given size and height. The main probability methods used in the book are the generalized allocation scheme and methods of the branching process theory. This monograph should be of value and interest to researchers in probabilistic methods and discrete mathematics.