The aim of the talk is to make a review of the recent advances of fractional calculus. New results and new trends will be stated.

In this talk we will make a survey on different methods for obtaining periodic orbits in dynamical systems. In particular we shall compare the classical combinatorial dynamics and the new methods coming from Algebraic Topology and Lefschetz fixed point theory.

We define new concepts of fractional quantum calculus by defining a new q-shifting operator. After giving the basic properties we define the q-derivative and q-integral. New definitions of Riemann-Liouville fractional q-integral and q-difference on an interval are given and their basic properties are discussed.

A model for cancer treatment is examined. The model is integrated into the Caputo-Fabrizio fractional derivative first, to examine the existence of the solution. Then, the uniqueness of the solution is investigated and we identified under which conditions the model provides a unique solution.