Summary: A general framework for the kinetic modelling of non-relativistic polyatomic gases is proposed, where each particle is characterized both by its velocity and by its internal state, and the Boltzmann collision operator involves suitably weighted integrals over the space of internal states. The description of the internal structure of a molecule is kept highly general, and this allows classical and semi-classical models, such as the monoatomic gas description, the continuous internal energy structure, and the description with discrete internal energy levels, to fit our framework. We prove the H-Theorem for the proposed kinetic equation of Boltzmann type in this general setting, and characterize the equilibrium Maxwellian distribution and the thermodynamic number of degrees of freedom. Euler equations are derived, as zero-order approximation in a suitable asymptotic expansion. In addition, within this general framework it is possible to build up new models, highly desirable for physical applications, where rotation and vibration are precisely described. Examples of models for the Hydrogen Fluoride gas are shown. A link between state-based and energy-based points of view is presented.