Uncertainty quantification (UQ) has been a hot research topic recently. UQ has a variety of applications, including hydrology, fluid mechanics, data assimilation, and weather forecasting.  Among others, high order numerical methods have become one of the most important tools for UQ; and the relevant computational techniques and their mathematical theory have attracted great attention in recent years. This talk will present a brief introduction to some recent developments on high order algorithms for both forward UQ and inverse UQ.