A Taylor series method for solving nonlinear structural dynamics problems where nonlinear items can be expressed as a polynomial with multiple variables was established. Different from existing methods, the Taylor series method satisfies governing equations in continuous intervals rather than at discrete time instants or in an average form. It solves dynamics problems through a sequence of recursions of Taylor expansion coefficients, without the necessity of solving simultaneous equations. The method was compared with the Runge-Kutta method through solving classical equations of Duffing, Van der Pol and the free vibration equation of two DOFs with quadratic nonlinear items. Numerical results indicated that the proposed Taylor series method is an excellent alternative for solving the aforementioned nonlinear dynamics problems.