In this paper the current status and open challenges of synthetic speech detection are addressed. The work comprises an initial analysis of available open datasets and of existing detection methods, a description of the requirements for new research datasets compliant with regulations and better representing real-case scenarios, and a discussion of the desired characteristics of future trustworthy detection methods in terms of both functional and non-functional requirements. Compared to other works, based on specific detection solutions or presenting single dataset of synthetic speeches, our paper is meant to orient future state-of-the-art research in the domain, to quickly lessen the current gap between synthesis and detection approaches.