This thesis examines the compactness in a variety of situations and forms. Thesis is divided into six chapters, and each chapter consists of several subsections. In the first chapter we introduce metric and topological spaces, their properties, and the very notion of compactness in metric and topological spaces, continuous function and its relationship with compactness. In the second chapter we study the convergent sequences and subsequences. We define accumulation points of sequences and totally bounded metric spaces and investigate their relationship with compactness. In the third chapter we find various types of compactness such as sequential compactness, compactness of subspaces and compactness in \(\mathbb{R}^n\). In this chapter, we also define a complete metric space. The fourth chapter deals with the application of the compactness and prove that a continuous function on a segment must be integrable. In the fifth chapter, we define a local compactness and metric space \((I^{\infty},d_{\infty})\). In the final, sixth chapter, we investigate compactness of Hilbert cube, and study a homeomorphism of metric and topological spaces.