Accurate prediction of the initiation and evolution of convective storms remains an important challenge in Numerical Weather Prediction. Until recently it was impossible to properly resolve any convection other than large organised convective systems and parametrization schemes were essential. The Met Office is now using a non-hydrostatic version of its Unified Model to develop a high-resolution forecast system for the future with a horizontal gridlength of ~1km. Ideally, such a system should be able to simulate the wide variety of convective events that are observed. The types of convection we see over the UK range from small-scale scattered showers that even a 1-km gridlength model is unable to resolve properly through to more organised clusters, frontal convection and deep summertime thunderstorms. Experiments have been performed to examine the behaviour of the high-resolution simulations of several very different convective situations. Results are presented from model forecasts using gridlengths of 12, 4, 2 and 1km. The simulations show how the initiation, intensity and organisation of convection can be very sensitive to resolution and to the characteristics of the convection scheme when included. An unphysical interaction between the dynamics and convection scheme can develop when a large amount of CAPE is present. The use of a modification to the convection scheme to limit the allowed mass flux and the sensitivity to other model parameters will also be discussed.