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We consider point sets in the real projective plane $\mathbb{R}P^2$ andexplore variants of classical extremal problems about planar point sets in thissetting, with a main focus on Erd\H{o}s--Szekeres-type problems.  We provide asymptotically tight bounds for a variant of theErd\H{o}s--Szekeres theorem about point sets in convex position in$\mathbb{R}P^2$, which was initiated by Harborth and M\"oller in 1994. Thenotion of convex position in $\mathbb{R}P^2$ agrees with the definition ofconvex sets introduced by Steinitz in 1913.  For $k \geq 3$, an (\affine) $k$-hole in a finite set $S \subseteq\mathbb{R}^2$ is a set of $k$ points from $S$ in convex position with no pointof $S$ in the interior of their convex hull. After introducing a new notion of$k$-holes for points sets from $\mathbb{R}P^2$, called projective $k$-holes, wefind arbitrarily large finite sets of points from $\mathbb{R}P^2$ with no\projective 8-holes, providing an analogue of a classical planar constructionby Horton from 1983. We also prove that they contain only quadratically many\projective $k$-holes for $k \leq 7$. On the other hand, we show that thenumber of $k$-holes can be substantially larger in~$\mathbb{R}P^2$ than in$\mathbb{R}^2$ by constructing, for every $k \in \{3,\dots,6\}$, sets of $n$points from $\mathbb{R}^2 \subset \mathbb{R}P^2$ with $\Omega(n^{3-3/5k})$\projective $k$-holes and only $O(n^2)$ \affine $k$-holes. Last but not least,we prove several other results, for example about projective holes in randompoint sets in $\mathbb{R}P^2$ and about some algorithmic aspects.  The study of extremal problems about point sets in $\mathbb{R}P^2$ opens anew area of research, which we support by posing several open problems.

Erd\H{o}s--Szekeres-type problems in the real projective plane