After stating the Riemann–Roch theorem and Serre duality for Enriques surfaces and making some elementary, yet useful observations, we turn to the vanishing theorems of cohomology on Enriques surfaces. On our way, we discuss ample, big, and nef invertible sheaves, as well as criteria to check whether a given invertible sheaf has these properties. Then, we turn to the Zariski decomposition, as well as general vanishing theorems.We refer to [611] for more on these notions for surfaces, and to [446] for a general background on positivity questions and applications. For vanishing theorems for surfaces over the complex numbers, we refer the interested reader to [43, Chap. 4.12].