By collaborative work the Sheffield Leicester Integrated Model for Microstructural Evolution in Rolling (SLIMMER) has been developed for hot rolling of flat products. The background physical metallurgy is presented together with the expressions used to describe microstructure evolution for a range of ferrous and non-ferrous metals. The finite difference thermal model at the heart of SLIMMER computes heat loss to air, descalers, rolls and water cooling while allowing for oxidation and deformation heating. The use of temperature compensated time enables isothermally determined equations for microstructure evolution to be applied to practical non-isothermal conditions. Rolling loads and torques are calculated using Sims theory with an accurate prediction of mean flow stress. Examples of rolling niobium microalloyed steel plate and the effect of initial grain size illustrate the capabilities of SLIMMER and show some of the validation of the predictions.