Ken yeah, it’s actually tricky question. We can’t easily answer on it without splitting the topic into smaller parts because PWA is just an idea which helps to handle few browser’s features.
http://caniuse.com/#feat=serviceworkers — iOS Safari and Opera Mini don’t support at all but support of Chrome for Android and Android Browser is enough
http://caniuse.com/#search=notification — almost same story like above
Web App Manifest
http://caniuse.com/#search=web%20app%20manifest — only Chrome for Android for now
Additionally check out this link: https://whatwebcando.today/ you can find out more about all features that you can (but don’t have to) use in your app for now.
These are the most important parts of PWA concept. However please notice that in Web Development you can always handle not-supported features in a good way (so it’s a working compromise). PWA is getting more and more popular, so we can expect better support among all mobile browsers.
Moreover, I recommend you to read this article: https://cloudfour.com/thinks/ios-doesnt-support-progressive-web-apps-so-what/ it explains why you should implement PWA anyway :-)