Time-scale calculus is a modern area of mathematics that unifies discrete and continuous analysis, more specifically the theories of differential equations and difference equations. It is done by constructing a theory for functions defined on so-called time scales which are nonempty, closed subsets of the set of real numbers. Time-scale calculus offers a very general setup to model time-dependent phenomena and thus has a high potential for applications. The purpose of the thesis is to discuss the fundamental concepts and ideas of time-scale calculus, including general description of time scales as well as differentiation and integration on time scales. In doing so, we investigate the properties and connection of the so-called delta and nabla derivatives and Henstock--Kurzweil delta and nabla integrals of functions defined on time scales.