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This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of (k,l)-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a (k,l)-algebraically stable two-step Runge-Kutta method with 0<k<1 is proved. For the convergence, the concepts of D-convergence, diagonally stable, and generalized stage order are firstly introduced; then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its generalized stage order is p, then the method with compound quadrature formula is D-convergent of order at least min{p,ν}, where ν depends on the compound quadrature formula.