The development of the study of sequence spaces got momentum by the introduction of new convergence methods. Some of them are statistical convergence, lacunary convergence, lacunary statistical convergence etc.In the thesis, I have basically concentrated on various type of convergence methods and developed new classes of statistical and lacunary convergent sequences by using Orlicz and Modulus functions. We have also introduced these convergence methods in the space of vector valued double difference sequences. The concept of statistical convergence which is defined for locally convex topological vector space, seminormed space etc. has been extended for the class of composite vector valued sequence spaces and further, we have proved some results analogues to the results obtained by earlier authors, like, Salat , Fridy , Connor  etc. By combining the concepts of matrix summability, lacunary convergence, Orlicz functions, difference sequences we obtained new class of sequence spaces which generalizes many known sequence spaces and also plays an important role in comparison to vector valued strongly Ces´aro-type summable sequence spaces.