In this monograph, a theoretical essay will be presented on the use of the Theory of Scalar Fields with a Classical approach in the study of Cosmology, since the cosmological models involving these fields allow the description of a phase of accelerated cosmic expansion and, therefore, present themselves as an alternative promising in the study of cosmic inflation and dark energy. Initially, the contents involved in the Theory of General Relativity, established in 1915 by Albert Einstein through Differential Geometry, which provides an indispensable mathematical tool for its formulation, will be addressed. Continuing, aspects of Standard Cosmology will be explored, where the main successes and problems of this cosmological model will be presented, and consequently, as a way to solve such problems, the topic of Inflation will be started, in which, in order to detail this term, it will be treated then, in a brief way, about the Dark Energy. Therefore, this course conclusion work seeks to develop, through the state of the art, a descriptive and introductory theoretical study of some of the main topics of Cosmology, thus establishing points of accessibility of these contents to undergraduate students, thus promoting , the dissemination of this theme to the academic community of the Mathematics and Physics course at ICET/UFAM.
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