This thesis is devoted to integrals and derivatives of arbitrary order and applications of the described methods in various fields. This study intends to increase the accessibility of fractional calculus by combining an introduction to the mathematics with a review of selected recent applications in physics. It is described general definitions of fractional derivatives. This definitions are compared with their advantages and disadvantages. Fractional calculus concerns the generalization of differentiation and integration to non-integer (fractional) orders. The subject has a long mathematical history being discussed for the first time already in the correspondence of G. W. Leibnitz around 1690. Over the centuries many mathematicians have built up a large body of mathematical knowledge on fractional integrals and derivatives. Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. In the first chapter, Grünwald-Letnikov approache to generalization of the notion of the differentation and integration are considered. In the second chapter, the Riemann Liouville definition is given and it is compared with Grünwald-Letnikov definition. The last chapter, Caputo.s definition is given. In appendices, two applications are given including tomography and solution of Bessel equation.