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In this paper a bijective correspondence between superminimal surfaces of anoriented Riemannian $4$-manifold and particular Lagrangian submanifolds of thetwistor space over the $4$-manifold is proven. More explicitly, for everysuperminimal surface a submanifold of the twistor space is constructed which isLagrangian for all the natural almost Hermitian structures on the twistorspace. The twistor fibration restricted to the constructed Lagrangian gives acircle bundle over the superminimal surface. Conversely, if a submanifold ofthe twistor space is Lagrangian for all the natural almost Hermitianstructures, then the Lagrangian projects to a superminimal surface and is iscontained in the Lagrangian constructed from this surface. In particular thisproduces many Lagrangian submanifolds of the twistor spaces $\mathbb{C} P^3$and $\mathbb{F}_{1,2}(\mathbb{C}^3)$ with respect to both the K\"{a}hlerstructure as well as the nearly K\"{a}hler structure. Moreover, it is shownthat these Lagrangian submanifolds are minimal submanifolds.

A note on Lagrangian submanifolds of twistor spaces and their relation to superminimal surfaces