This chapter defines statistical hypothesis tests mathematically. Those tests assume two sets of probability models, called the null and alternative hypotheses. A decision rule is a function that depends on the data and a specified ?-level and determines whether or not to reject the null hypothesis. We define concepts related to properties of hypothesis tests such as Type I and II error rates, validity, size, power, invariance, and robustness. The definitions are general but are explained with examples such as testing a binomial parameter, or Wilcoxon–Mann–Whitney tests. P-values are defined as the smallest ?-level for observed data for which we would reject the null at that level and all larger levels. Confidence sets and confidence intervals are defined in relation to a series of hypothesis tests with changing null hypotheses. Compatibility between p-value functions and confidence intervals is defined, and an example with Fisher’s exact test shows that compatibility is not always present for some common tests.