  We develop continuous-stage Runge-Kutta methods based on weighted orthogonal
polynomials in this paper. There are two main highlighted merits for developing
such methods: Firstly, we do not need to study the tedious solution of
multi-variable nonlinear algebraic equations associated with order conditions;
Secondly, the well-known weighted interpolatory quadrature theory appeared in
every numerical analysis textbook can be directly and conveniently used. By
introducing weight function, various orthogonal polynomials can be used in the
construction of Runge-Kutta-type methods. It turns out that new families of
Runge-Kutta-type methods with special properties (e.g., symplectic, symmetric
etc.) can be constructed in batches, and hopefully it may produce new
applications in numerical ordinary differential equations.
