ASP.NET already runs on .NET with Mono, but not always the latest versions of things. Having it all open source could bridge that compatibility gap and let us fully enjoy the ecosystem, instead of having to be careful with taking dependencies all over.
ASP.NET support on Mono is very shaky, with random things missing. I tried to develop a Mono ASP.NET MVC app on it recently and I had to switch to Windows due to the number of issues.
For webapps, I am using Nancy and .Net 4.5 and I am very happy with it. Developing debugging on windows and deploying it on ubuntu apache with mod mono works great for me.
ASP.NET is a big framework and indeed it's never been as smooth as it could be, especially with the very latest versions. It's not to say that web dev is not possible with Mono. I've been using Owin+Nancy+Signalr successfully on Linux.
