Abstarct: The study of relativistic particles in space-time of topological defects has been developing research field. The analysis of the influence of the spacetime produced by topological defects on the energy spectrum of the particles, has been considered under both non- relativistic and relativistic points of view. According to the modern concepts of theoretical physics, different types of topological defects may be produced by the vacuum phase transition in the early Universe. These include domain walls, cosmic strings and monopoles. Among them, cosmic strings seem to be the best candidates to be considered and is one of the most important examples of topological defects. Recently, several examples of studies in a space-time with a cosmic string were done revealing the role played by this topological defect in curved space-time. Our intention now is to extend these studies to consider this dynamics in general background of curved space time. So we introduce different curved space time such as cosmic string, Fermi-walker, spining cosmic string, space time with noninertial effect, space time with spherical symmetry. We consider the effect of these space time on both non- relativistic and relativistic particles described by Dirac equation for spin half particles, DKP equation for spin 0 and 1 particles in these kind of curved space-time.