Quantum impurity system（QIS）is one of the fundamental problems in theoriti-cal physics,where system properties depend on the strong e-e interactions among localized impurities.Correct characterization of QIS leads us to understand the mechanism of strong electron correlation.However,this has remained rather a chal-lenging problem because such an impurity has internal degrees of freedom（intrinsic angular momentum or spin）which interacts with the surrounding electrons.As a result,it becomes a challenging many body problem.In the last three decades,different types of techniques have been established to achieve an accurate characterization of QIS.Despite the success of these methods,the feasibility of these methods has been restricted by their limited efficiency and correctness.Moreover,the applicability of these methods is limited to few basic models because with an increase in the system size,the numerical complexity of these methods also increases.As a result,it is difficult to generalize these methods to more complex models.Consequently,a universal and more efficient method is needed to accurately address strong correlation effects in general quantum impurity systems.In this thesis,we present the quantum dissipative theory（QDT）to theoretically address QIS.In QDT,the whole model is usually divided into two parts,the impu-rity is regarded as a system and the surrounding environment is treated as a bath or reservoir which usually possesses infinite degrees of freedom.Thanks to path-integral formulation,the influence of the bath on the system can be captured in few stochastic auxiliary fields.In the case of bosonic bath,these auxiliary fields are realized by c-numbers which can be represented by conventional means.Unlike bosonic baths,the auxiliary fields for fermionic baths are realized by Grassmann numbers,which mutually anti-commute.Grassmann numbers can not be repre-sented by conventional means.This difficulty restricted the early attempts to only formal derivations without any promising feasible numerical simulations.Recently,our research group proposed a mapping scheme where Grassmann numbers can be mapped into c numbers（white noises）and some pseudo ladder operators.This map-ping scheme results in a very feasible and accurate stochastic equation of motion（SEOM）for the reduced dynamics of a system connected to a fermionic bath.In this thesis,we are presenting a very detail numerical and technical discussion on the whole formalism including thorough explanation of the mapping of Grassmann numbers into c-numbers.Moreover various key quantities,i.e.,electron occupation number and the current flow from the system into the attached reservoirs are also calculated.This thesis can be divided into three major parts.The first part is an introduc-tion to stochastic formalism.It covers the construction of stochastic theory for the system connected to a bosonic bath.Furthermore,it also includes a section on the numerical treatment of stochastic differential equations.In addition,it introduces us to Grassmann algebra and Complex algebra.The second part of the thesis is related to the construction of stochastic formalism for a system connected to a fermionic bath.The third part belongs to the numerical treatment and results.This thesis consists of three chapters and detail about each chapter is given below.Chapter 1 is an introductory chapter.It introduces the reader to open quantum system and quantum dissipative theory（QDT）.The chapter starts with the discus-sion on Brownian motion and leads us to Einstein s Brownian motion theory and Langevin’s stochastic approach.As Langevin’s approach has its own limitations and can not be generalized to open quantum dynamics,in this chapter,we show that Feynman-Vernon integral formulation can be used to successfully decouple system from the bosonic bath.Feynmann-Vernon decoupling technique results in a very useful stochastic differential equation（SDE）for the reduced dynamics of the system.After all the introduction and derivations,the numerical treatment of SDEs is done in detail.As the considered SDE contains color noises,we present different meth-ods along with explicit algorithms for the generation of color noises.Moreover,the chapter covers a section on the correspondence between deterministic hierarchical equations of motion（HEOM）and stochastic equation of motion.At the end,the chapter introduce us to complex algebra and Grassmann algebra.In chapter 2,we derive a stochastic equation of motion for the dissipative dynam-ics of a system connected to a fermionic bath.The many body effects are taken into account by using Grassmann random fields.Since Grassmann numbers are nu-merically unfeasible,we map them into feasible c-numbers and some pseudo ladder operators which belong to an auxiliary space.The mapping is incomplete but still the resultant SDE can do exact characterization of non-interacting systems and can give approximate results for the interacting systems.Chapter 3 covers a detail numerical and technical discussion on stochastic equation of motion derived in chapter 2.In this chapter,we present explicit algorithms for the numerical representation of the system and reservoir operators.The applicability of SEOM is demonstrated on the two level system at both high and low temperatures along with benchmark results for the various key quantities,i.e.,electron occupation number and the current flow from the system into attached reservoirs.Moreover,we include a detail discussion on the numerical stability and convergence of our SEOM.Chapter 4 is devoted to thesis conclusion and perspective.