The topic of this thesis is vanishing theorems in positive characteristic. In particular, we use "the covering trick of Ekedahl" to investigate the vanishing of $H^1(X, \mathcal{O}_X(-D))$ for a big and nef Weil divisor $D$ on a normal projective variety with $-K_X$ nef. In dimension two, we show that on a surface of log del Pezzo type over a perfect field of characteristic $p>5$ this vanishing holds. More generally, using techniques of the \emph{Minimal model program} we prove the Kawamata--Viehweg vanishing theorem in this setting. We also construct a counter-example in characteristic five, showing that our result is optimal. We discuss the relationship (due to Hacon--Witaszek) between this vanishing theorem and properties of threefold klt-singularities. We investigate if a similar relationship exists between threefold lc-singularities and a certain vanishing theorem for higher direct images of elliptic fibrations. This leads to a counter-example to a theorem of Koll'ar over the complex numbers, in every positive characteristic.