Automorphism groups of $P^1$-bundles over a non-uniruled base

In this survey we discuss holomorphic ${\mathbb{P}}^1$-bundles $p:X\to Y$ over a non-uniruled complex compact K"ahler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We discuss so called Jordan properties of the groups $Aut(X)$ and $Bim(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively.