I will present recent work in collaboration with Simone Murro and Matteo Capoferri on some foundational aspects of noncommutative derived geometry. The talk is intended for a non-specialist audience and will begin with an overview of the history of the problem of constructing the spectrum of a noncommutative ring. I will then argue that, within the framework of derived geometry, the problem admits a natural and simple solution. In the final part of the talk, I will describe the outcome of our construction and provide examples, time permitting.