We present a multi-agent logic of belief and announcements wherein the sending of announcements and the reception of announcements by agents are separated, thus straying from the paradigm of Public Announcement Logic (PAL). Both PAL and Asynchronous Announcement Logic (recently proposed in the literature) are special cases in our framework. We provide a history-based semantics for our 'Partially Synchronous Announcement Logic', proposing three different interpretations of the notion of asynchronic-ity. We then show that the logic of our three proposals is the same ('PSAL') and prove soundness and completeness for a Hilbert-style axiomatisation. Finally, we propose a notion of common belief for this framework, of which we give some validities.