Abstarct: Heun functions are solutions of the Heun differential equation which is a generalization of the hypergeometric equation. The Heun differential equation is the most general second-order equation having four regular singular points that in recent years it has attracted considerable attention. Now, this differential equation appears in many fields such as particle physics, nuclear physics, quantum mechanics and is used to describe a wide range of physical phenomena. Heun differential equation in the study of relativistic and non-relativistic particles appears for many potentials and its eigenfunctions are defined in terms of Heun functions. One of the important applications of the Heun equation is in the general relativity and cosmology. In this research we introduce Heun function and their confluent forms, then we study this function in the relativistic systems by Dirac equation for spin half particles, Weyl equation for spin half massless particles, Klein- Gordon equation and DKP equation for spin 0 particles. Today, the study of relativistic particles in space-times created by topological defects has attracted much attention. A cosmic string is one of the most important examples of topological defects. In this research, we introduce different space-times such as cosmic string, spinning cosmic string, space-time with noninertial effects and Gödel-type space-times and we study the influence of topological defect on the energy spectrum of relativistic particles. We also study the Weyl equation in the presence of variable magnetic fields and investigate the influence of these fields on the energy spectrum of particles. The purpose of calculating energy is to obtain the thermodynamic quantities of the system.