In addition to this more "official" work on getting .NET and WebAssembly to play nicely, there is another effort to get a small .NET runtime working in the browser [1]. The demos are compelling [2].

[1] https://github.com/SteveSanderson/Blazor / https://github.com/ncave/dotnet-js

[2] https://youtu.be/MiLAE6HMr10