Same-day delivery has always been possible through specialized (and expensive) courier service, but now major established delivery as well as several startup companies are getting into the act.
FedEx just announced it has expanded same-day delivery service to 1,800 cities. The deliveries are made by FedEx drivers in FedEx trucks (in other words, they aren't outsourced to other carriers), and the service includes a mobile app for tracking.
Deliv is an example of a startup company offering same-day e-commerce delivery. Deliv is present in 19 markets, and Walmart's purchase of Parcel gives it the ability to same-day deliver in New York City.
With the emphasis now on speed-of-delivery, expect these companies and more (including, of course, Amazon) to continue to improve their delivery systems, making them as fast and inexpensive as possible.