The main topic of this book is a construction of a Fukaya category, an object capturing information on Lagrangian submanifolds of a given symplectic manifold. Fukaya categories are of interest due to the recent formulation of homological mirror symmetry. The first part of the book is a self-contained exposition of A∞-categories and the underlying homological and homotopical algebra. In the second part, the actual construction of a Fukaya category is presented. The author first presents the main ideas by giving a preliminary construction and then he proceeds in greater generality, though the complete generality already present in recent literature is not reached. The last part treats Lefschetz fibrations and their Fukaya categories and briefly illustrates the theory on the example of Am-type Milnor fibres. The book is written in an austere style and references for more detailed literature are given whenever needed. The reader is expected to have a certain background in symplectic geometry.