It is now late 2018 and it is looking grim honestly.
CefSharp, while under active development and has somewhat good documentation, still does not support .NET Core or Mono, rendering it unusable for anything .NET Core, .NET Standard, or Mono related (we exclusively use Linux for production servers so this is a hard no, sadly).
CefGlue, in comparison, apparently does support .NET Core/Mono according to the other posters. However, and this is important, the project moved to GitLab and does not have any license attached to it. The website is also unreachable (at least the one I found) and documentation is nonexistent.
There is a fork here which seems to be inactive again though. There is also an unofficial port to .NET Core here, but that seems to have a hard dependency on Avalonia.
However, there is Chromely, which seems to support cross platform and is based on CefSharp and CefGlue (and unofficial ports of those). It seems like that's a fully fledged browser though and not a library for embedding something in your application.
Since I was initially looking for an easy way to embed a headless browser in my application for crawling purposes (eliminating the need to install Chrome), I also took a look at Awesomium, but they seem to have moved to a new project called "Ultralight", which does not support C# and has no bindings for it at the moment.
As a last effort, there seems to be Optimus, which seems to support .NET Standard and is a full implementation of a WebBrowser in .NET, minus GUI. I'll try that out and maybe edit this answer if it works (or not).