This paper presents likelihood-based statistical tests for discovering new phenomena and constructing confidence intervals in high energy physics. The authors focus on methods that account for systematic uncertainties and derive asymptotic distributions of test statistics using results from Wilks and Wald. They introduce the concept of an "Asimov data set," a representative data set used to estimate the median experimental sensitivity of a search or measurement. The paper outlines the formalism of a search as a statistical test, defines test statistics such as $ t_{\mu} $, $ \tilde{t}_{\mu} $, $ q_{0} $, and $ q_{\mu} $, and discusses their distributions under the assumption of the Wald approximation. The authors provide formulas for the asymptotic distributions of these test statistics, which allow for the calculation of p-values and significance levels without the need for Monte Carlo simulations. The paper also discusses the use of these statistics in determining upper limits and confidence intervals for signal parameters, and highlights the importance of considering the median significance when interpreting results. The methods are applied to various examples, and the paper concludes with a discussion of the limitations and practical implications of the asymptotic approximations.This paper presents likelihood-based statistical tests for discovering new phenomena and constructing confidence intervals in high energy physics. The authors focus on methods that account for systematic uncertainties and derive asymptotic distributions of test statistics using results from Wilks and Wald. They introduce the concept of an "Asimov data set," a representative data set used to estimate the median experimental sensitivity of a search or measurement. The paper outlines the formalism of a search as a statistical test, defines test statistics such as $ t_{\mu} $, $ \tilde{t}_{\mu} $, $ q_{0} $, and $ q_{\mu} $, and discusses their distributions under the assumption of the Wald approximation. The authors provide formulas for the asymptotic distributions of these test statistics, which allow for the calculation of p-values and significance levels without the need for Monte Carlo simulations. The paper also discusses the use of these statistics in determining upper limits and confidence intervals for signal parameters, and highlights the importance of considering the median significance when interpreting results. The methods are applied to various examples, and the paper concludes with a discussion of the limitations and practical implications of the asymptotic approximations.