VB.NET on Linux
There are a few, like SimpleBasic, GnomeBasic and XBasic. None of them are fully compatible with Visual Basic.
The above answer was accepted eons ago, but is horribly outdated, since more recently, there's also .NET Core. This will run the actual VB.NET language, but it will not use Windows Forms controls and features powering most real VB.NET applications. .NET Core 3 does support some variation of Windows Forms, but only on Windows.
Please check Pedro Polonia's excellent answer that contains all the details that mine misses.
Is Visual Basic supported by .NET Core on Linux?
No, there is no Visual Basic support yet. At the very bottom of the roadmap in the "future work" section, it said at one time, that Visual Basic support was set for quarter 3 of 2016, but it is still not released as of March 2017. It is still planned.
Since ASP.NET Core is the most common thing that targets the .NET Core framework, the answer I've provided refers to that.
Converter for VB.NET Code to Linux Platform Exist?
Here's a link to the Mono Migration Analyzer to get started. It will help you pinpoint Microsoft specific calls, but you'll probably have to do the db conversion and data access layer manually. You may be surprised - mono does have a System.Data.SqlClient namespace so you may not have much work to do.
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