The last decades have seen a tremendous extension of the Brunn-Minkowski theory of convex bodies, to \(L_p\), Orlicz, and dual versions. Here, the Orlicz projection body is introduced. Let \(\phi:{\mathbb R}\to[0,\infty)\) be a convex function with \(\phi(0)=0\), strictly decreasing on \((-\infty,0]\) or strictly increasing on \([0,\infty)\). For a convex body \(K\subset {\mathbb R}^n\), the support function of the Orlicz projection body \(\Pi_\phi K\) is defined by \[ h(\Pi_\phi K,x)=\inf\left\{\lambda>0: \int_{\partial K} \phi\left(\frac{\langle x,\nu(y)\rangle}{\lambda\langle y,\nu(y)\rangle}\right)\langle y,\nu(y)\rangle\,d{\mathcal H}^{n-1}(x)\le n|K|\right\}\] for \(x\in{\mathbb R}^n\). Here \(|K|\) denotes the volume of \(K\), \(\langle\cdot\,,\cdot\rangle\) is the scalar product, \({\mathcal H}^{n-1}\) the \((n-1)\)-dimensional Hausdorff measure, and \(\nu\) the (\({\mathcal H}^{n-1}\)-almost everywhere on \(\partial K\) unique) outer unit normal vector of \(K\). For \(\phi(t)=|t|\), this yields the classical projection body (introduced by Minkowski), up to a factor. Several later generalizations are included. If \(A\) is a volume preserving linear transformation, then \(\Pi_\phi AK= A^{-t}\Pi_\phi K\). The main result of the paper is the Orlicz-Petty projection inequality, saying that, for \(K\) containing the origin in the interior, the volume quotient \(|\Pi_\phi^\circ|/|K|\), where \(^\circ\) indicates the polar body, is maximized when \(K\) is an ellipsoid with center at the origin. That this is the only extremal, is proved if \(\phi\) is strictly convex, and was later proved without this assumption by K. Böröczky jun. [J. Differ. Geom. 95, No. 2, 215–247 (2013; Zbl 1291.52012)], who also obtained corresponding stability results.