We give a brief survey on the theory of symplectic connections over a symplectic manifold $(M,\omega)$, ie a torsion free linear connection such that $\nabla \omega=0$. A result of P. Tondeur assures that these connections always exist. Some unsolved problems still arise in the case of $(\mathbb{R}^{2n},\omega)$ and thus we look carefully on a new result for translation invariant connections.

Next we introduce the bundle of all compatible complex structures on the tangent spaces of $M$, which is called a Twistor Space if it is seen together with a canonical almost complex structure $J^\nabla$ induced by a symplectic connection on $M$. The integrability equation of $J^\nabla$ appears in terms of the curvature: it agrees with the vanishing of the Weyl curvature tensor. With this theorem, which in a final correct version is due to J. Rawnsley, we may draw a bridge between the study of symplectic connections and that of twistor spaces. We proceed with the presentation of a certain biholomorphism of the twistor space induced by some symplectomorphism on the base manifold. This is used in understanding farther the action of $\operatorname{Symp}(M,\omega)$ on the space of symplectic connections. We give examples. Finally and having time, we show some complex geometrical properties of the twistor space, namely a kählerian structure if and only if $\nabla$ is flat, the degree of holomorphic completeness and the vanishing of the Penrose transform in the symplectic setting.

Notice that the referred result on the Weyl tensor strongly resembles the analogue of twistor theory in riemannian geometry. Hence we take the opportunity of this seminar to make a commemoration of the 25 years of that landmark in twistors which is Self-duality in four-dimensional riemannian geometry from M. Atiyah, N. Hitchin and I. Singer.