Abstarct: In this thesis, fractional variable order discrete state-space systems baxsed on different definitions of fractional variable order difference are investigated. The general solution of these systems is derived. Moreover, necessary and sufficient conditions for reachability and observability are given and proven. The sufficient conditions for controllability are proposed too. Before this, in previous chapters, we intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. The fractional variational calculus is a recent mathematical field that consists in minimizing or maximizing functionals that depend on fractional operators (integrals or/and derivatives). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Chapter 2 presents a short introduction to the classical calculus of variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the fractional calculus of variations and some fractional variational problems with variable-order. In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove fractional Euler–Lagrange equations for several types of fractional problems of the calculus of variations, with or without constraints. In last chapter, we study fractional variable order discrete state-space systems baxsed on different definitions of fractional variable order difference.