Great overview by Maximiliano Firtman on progressive web apps, which are now supported on iOS 11.3 thanks to new web technologies Apple adopted in Safari 11.1.
I wouldn’t call progressive web apps a “replacement” for native software from the App Store (just read the list of technical limitations from Firtman’s article), but they are indeed a remarkable improvement over how Safari used to save web apps to the Home screen. I recommend checking out these two lists to try some progressive web apps and see how they work on iOS. I spent way too much time playing around with Web Flap, a Flappy Bird clone that runs as a progressive web app and even supports offline mode on iOS 11.3.