What is this method of searching for flights called and do any sites still offer it?

My boss told me that he once got a cheap ticket to Italy the next day by typing in the date and selecting the destination after the fact based on price and desirability. (Similar to just going to the airport and choosing a departing flight, I guess.)

I’ve always wanted to do something spontaneous like this but can’t find any sites that offer it. Does anyone know any that still do? Have you tried it?