The Mukai conjecture is a numerical characterisation of products of projective space among smooth Fano varieties in terms of the Picard number, Fano index (or pseudo-index), and dimension. I will discuss two cases where the Mukai conjecture has been proved.

The first case is for the spherical varieties: the natural generalisation of toric varieties to the setting of reductive groups. Here the Mukai conjecture is proved using a connection to the complexity of log Calabi-Yau pairs: a birational criterion detecting toric varieties.

The second case is motivated by the fact that given a Cox ring presentation, a Mori dream space X embeds canonically into a toric variety Z. We ask when the Mukai conjecture for a Fano variety X can be "inherited" from Fujita's log version of the Mukai conjecture for Z. I will explain how under suitable assumptions on the Cox ring this approach can prove the Mukai conjecture for X.