This answer is is obsolete. Microsoft has released .NET for Linux.

Please see: https://www.linode.com/community/questions/274

The answer, like many, is: It depends.

If you are using Perl as your scripting language, Apache::ASP will allow you to use ASP on Linux but you may need to make some changes to make sure it works correctly in the Linux environment.

Their FAQ is available here.

There was commercial implementation called Chili!Soft ASP available at one time, but this was purchased by Sun Microsystems which in turn was purchased by Oracle and does not appear to be available as a separate product anymore. (Going to the original domain chilisoft.com, redirects you to Oracle, and searching Oracle for Chili!Soft ASP does not return any hits.)

If your ASP application is based on ASP.NET, you may be able to run it using Mono. It's FAQ page says it does not support old style ASP pages, but mentions a tool that can assist with translating your code.