Abstract: The theory of adjoint operators is widely used in solving applied multidimensional problems with the Monte Carlo method. Efficient algorithms are constructed using the duality principle for many problems described in linear integral equations of the second kind. On the other hand, important applications of adjoint equations for designing experiments were suggested by G.I. Marchuk and his colleagues in their respective works. Some results obtained in these fields are also generalized to the case of nonlinear operators. Linearization methods are mostly used for that purpose. The results for Lyapunov–Schmidt nonlinear polynomial equations are obtained in the theory of Monte Carlo methods. However, many interesting questions in this subject area remain open. New results about dual processes used for solving polynomial equations with the Monte Carlo method are presented. In particular, the adjoint Markov process for the branching process and corresponding unbiased estimate of the functional of the solution to the equation are constructed in the general form. The possibility of constructing an adjoint operator to a nonlinear one is discussed.