.NET Core is the equivalent to mono. However the .NET Core libraries were a bit painful to port to since they were limited. So they are expanding this with .NET Standard - which is an API specification - which you can use and work across all .NET platforms (.NET Framework, .NET Core, or Xamarin).

Keep in mind it sounds like .NET Standard basically is between the full .NET Framework and .NET Core. So if you have an existing .NET Framework app it's very possible it would just work on .NET Standard.