This survey explores the interplay between twistor geometry and projective geometry, focusing on their applications to algebraic surfaces. We explore two main topics: the inclusion of twistor fibers and lines in these surfaces, and the behavior of twistor discriminant loci, with a particular focus on degree-2 surfaces. The study highlights contributions from Ballico and collaborators, comparing the twistor spaces of \({\mathbb{C}\mathbb{P}}^3\) and the flag threefold \({\mathbb {F}},\) which reveal fascinating contrasts and parallels. Key findings include a detailed analysis of surfaces in \({\mathbb{C}\mathbb{P}}^3\) and \({\mathbb {F}}\) that either contain finite or infinite twistor fibers. The survey also touches on cubic surfaces in \({\mathbb{C}\mathbb{P}}^3\) and their counterparts in \({\mathbb {F}},\) where the configurations of twistor fibers lead to intriguing results. Special attention is given to surfaces of twistor degree 2, including how their geometry and singularities interact with twistor projections. In particular, we discuss smooth and singular surfaces in \({\mathbb {F}}\) of bidegree (1, 1) and (0, 2), as well as their discriminant loci.
