Fractional calculus, a popular name for the calculus of arbitrary order, is a branch of mathematical analysis which basically studies several different ways of defining a derivative. With these different definitions, it is possible, for instance, to build and solve fractional differential equations. We present here a short introduction to the subject. Specifically, we discuss two types of fractional derivatives, the Riemann-Liouville and the Caputo fractional derivatives. We also introduce the one-parameter Mittag-Leffler function, the queen of special functions of fractional calculus, together with two generalizations, also called Mittag-Leffler functions, with two and three parameters. We use integral transforms to solve simple fractional ordinary differential equations and simple fractional integral equations. Some solved exercises are discussed step-by-step, among which the Laplace transform of the Prabhakar function. As an application we study the fractional partial differential equations associated with relaxation and oscillation. We conclude with a list of proposed exercises; Some interesting applications are left to the reader, such as the Christoffel-Darboux formula, all of them with the answer and/or suggestion.