The project is aimed at the study of quantum aspects of extended supersymmetric field theories with various number of supersymmetries in three and four space-time dimensions. In particular we will develop extended superfield approaches to superconformal gauge theories with N=4 supersymmetry in four dimensional space-time and to supersymmetric Chern-Simons-matter theories with N=3,4,6,8 supersymmetries in three-dimensional space-time. Various classical and quantum aspects of these theories such as the construction of quantum effective action and computation of correlation functions of composite operators will be considered by using the developed extended superspace methods. These problems are motivated by the development of superstring / M-theory which is considered to be a candidate for the unified theory of all fundamental interactions of elementary particles. In particular, the four-dimensional N=4 supergauge theories effectively describe systems of D3 and M5 branes in supergravity and M-theory while three-dimensional N=6 and N=8 supergauge theories correspond to M-theory 2-branes. This gives rise to different manifestations of so-called AdS/CFT correspondence (or field theory / gravity correspondence). Within this project we will explore the properties of classical and quantum dynamics of above mentioned extended supersymmetric field theories that will shed new light on various aspects of the field theory / gravity correspondence.