Scalar field cosmology Pt. I: A canonical scalar field in cosmology pt. II. Advanced methods of exact solutions construction pt. III. Cosmological perturbations pt. IV. Friedmann vs Abel equations: a connection pt. V. The phantom fields pt. VI. Branes. This monograph discusses cosmological inflation and provides exact and slow roll solutions. It also reviews new and advanced approaches of exact solutions construction with canonical scalar fields including application of generating functions methods the superpotential and many others. This book presents the reduction of the Friedmann equation to the Abel equation which is a very useful tool in cosmology. It offers new solutions and discusses its properties. Additionally it touches upon the role of phantom scalar field cosmology and analyzes phantonical models. It describes brane cosmology with scalar fields providing exact solutions construction using the superpotential method as well as Darboux transformations. This book provides detailed calculations throughout.