This paper develops the mathematical framework and the solution of a system of type-2 fuzzy ordinary differential equations (T2FDE). The theory of T2FDEs is developed with type-2 fuzzy initial values, type-2 fuzzy boundary values and type-2 fuzzy parameters. Some natural phenomena can be modelled as dynamical systems whose initial conditions and/or parameters may be imprecise in nature. The imprecision of initial values and/or parameters is generally modelled by fuzzy sets. Here, the concept of generalized H2-differentiability, based on the extension of the class of differentiable type-2 fuzzy mappings or Hukuhara derivatives, is applied. Numerical simulations of type-2 fuzzy differential equations have also been developed. Several illustrative examples have been provided for different T2FDE models related to environmental dynamics and problems in mathematical biology.