In this thesis we introduce the variable exponent bounded variation spaces in the Riesz sense, we prove some embedding results and show that a type of Riesz representation lemma is valid in the newly introduced spaces. On the other hand, we characterize global Lipschitz Nemytskii operators defined in the Riesz bounded variation space with variable exponent. Finally we characterize the linear functionals on variable exponent Bochner-Lebesgue spaces in terms of the variable exponent Riesz bounded variation spaces for vector measures.