After extensive efforts over the course of a decade, convective–scale weather forecasts with horizontal grid spacings of 1–5 km are now operational at national weather services around the world, accompanied by ensemble prediction systems (EPSs). However, though already operational, the capacity of forecasts for this scale is still to be fully exploited by overcoming the fundamental difficulty in prediction: the fully three–dimensional and turbulent nature of the atmosphere. The prediction of this scale is totally different from that of the synoptic scale (10^3 km) with slowly–evolving semi–geostrophic dynamics and relatively long predictability on the order of a few days. Even theoretically, very little is understood about the convective scale compared to our extensive knowledge of the synoptic-scale weather regime as a partial–differential equation system, as well as in terms of the fluid mechanics, predictability, uncertainties, and stochasticity. Furthermore, there is a requirement for a drastic modification of data assimilation methodologies, physics (e.g., microphysics), parameterizations, as well as the numerics for use at the convective scale. We need to focus on more fundamental theoretical issues: the Liouville principle and Bayesian probability for probabilistic forecasts; and more fundamental turbulence research to provide robust numerics for the full variety of turbulent flows. The present essay reviews those basic theoretical challenges as comprehensibly as possible. The breadth of the problems that we face is a challenge in itself: an attempt to reduce these into a single critical agenda should be avoided. Scientific challenges of convective-scale numerical weather prediction