  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.
