DifferentiAlequations are powerful tools to model the time evolution of dynamical　　 systems, which have arisen widely in mechanics, physics, biology, ecology and the other　　 scientific fields. In the reAlworld, dynamicAlsystems are usually in?uenced by delay phe-　　 nomenon or random perturbation. Therefore, differentiAlequations containing these factors　　 can better simulate the reAldynamicAlsystems. However, the increasing complexity of　　 equations makes it almost impossible to get their explicit solutions, and hard to study their　　 dynamicAlproperties. Thus, it is of great significance to develop numericAlmethods for　　 solving these differentiAlequations, and analyze the properties of numericAlsolutions, such　　 as convergence and stability. In this thesis, we focus on severAltypes of differentiAlequa-　　 tions including stochastic and delay arguments, develop one-step methods for solving these　　 equations, and study the convergence and stability of these methods.　　 In chapter 2, by combining the idea of split-step methods with stochastic θ-methods, we　　 propose a class of split-step θ-methods for solving stochastic delay differentiAlequations. It　　 is proved that the methods are convergent with strong order 0.5 in the mean-square sense.　　 Chapter 3 is devoted to study the index-1stochastic delay differential-algebraic equa-　　 tions (SDDAEs) of retarded type. The existence and uniqueness of strong solutions are　　 derived under uniform Lipschitz condition and some generAlassumptions.　　 In chapter 4, we develop a generAlframework for a class of one-step methods for　　 solving index-1SDDAEs of retarded type, and establish the strong convergence criterion.　　 Based on the criterion, we construct some specific schemes for solving index-1SDDAEs of　　 retarded type and semi-explicit stochastic differential-algebraic equations of index-1.　　 In chapter 5, by adapting the existed numericAlmethods of ordinary differentiAlequa-　　 tions, a class of extended Rosenbrock numericAlsimulation methods for solving discrete-　　 distributed delay systems of neutrAltype are constructed, and some criteria, for judging　　 that the numericAlmethods are asymptotically stable, are obtained. It is shown that meth-　　 ods extended from A-stable classicAlRosenbrock methods can preserve the asymptotically　　 stability of the underlying linear system.　　 In chapter 6, for a class of nonlinear functional-integro-differentiAlequations, a type　　 of mixed Runge-Kutta methods are presented by combining the underlying Runge-Kutta　　 methods and the compound quadrature rules. Based on the non-classicAlLipschitz condi-　　 tion, some globAland asymptoticAlstability criteria are derived for the methods. Several　　 specific mixed Runge-Kutta methods of high precision and good stability are derived.　　 With numericAlexperiments, efficiency of the proposed methods and applicability of　　 the theoreticAlresults are further illustrated.