I shall talk about what happens when an entire sample path of a Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high level is exceeded, the conditional shape of the process above the high level, and the location of the minimum of the process given that the sample path is above a high level. The method of answering the questions are different for smooth and non-smooth processes. The former case is analysed in much more details than the latter.