This chapter is dedicated to the introduction of the Lebesgue integral, which is an object where the integrand is a measurable function and the integrator is a measure. It is defined in three steps: First for integrands which are simple nonnegative functions, then for nonnegative measurable functions, and eventually for all measurable functions for which the integral exists. Basic properties of the Lebesgue integral are shown. The lemma of Fatou for functions, the monotone convergence theorem, and Lebesgue’s theorem on dominated convergence are proven which provide conditions when limits and the Lebesgue integral may be interchanged. Lebesgue’s criterion for Riemann integrability provides a practical criterion under which conditions the integral with respect to the Lebesgue measure coincides with the Riemann integral. The chapter continues with the change of variable formula, theorems of Tonelli and Fubini, and the inequalities of Markov, Jensen, Hölder, and Minkowski. The chapter ends with the proof of Hoeffding’s inequality, Wald’s identity, and a short excursion to atomless probability spaces.