Isotrivially fibred surfaces and their numerical invariants

We give a survey of our previous work on relatively minimal isotrivial fibrations $\alpha \text{\hspace{0.17em}⁣}:X\to C$, where $X$ is a smooth, projective surface and $C$ is a curve. In particular, we consider two inequalities involving the numerical invariants $K_X^2$ and $\chi ({\mathcal{O}}_X)$ and we illustrate them by means of several examples and counterexamples.