(This answer is regularly updated whenever a new .NET announcement is made, and it has been updated with details of ASP.NET Core 3.0, .NET 3, and .NET 5)

Microsoft needs a good slapping for the amount of confusion over the past 3 years 5 years over .NET Core / DNX, ASP.NET Core, .NET Standard, .NET 5 and the rest.

(and I’m saying that as a former FTE SE in DevDiv… _{(“hi from building 16!”)})

• ASP.NET aka “System.Web” is now dead. WebForms is dead (hurrah!)
• ASP.NET MVC launched in 2008 was built on-top of ASP.NET, but bypassed most of the WebForms infrastructure.
  • ASP.NET MVC has its own versioning separate from ASP.NET and ASP.NET Web API (and ASP.NET Core). You linked to ASP.NET MVC 5‘s documentation – this is not the same thing as ASP.NET 5.
• ASP.NET Web API launched in 2012 is a sibling of ASP.NET MVC, in that it built on-top of ASP.NET too, but had its own class library (System.Web.Http) that didn’t share much with ASP.NET MVC (System.Web.Mvc). Attempting to combine an ASP.NET Web API service with an ASP.NET MVC web-application in the same project is an exercise in pain.
• ASP.NET MVC 5 was launched in 2014 as an update to ASP.NET MVC 4. It is unrelated to ASP.NET 5.
  • I do note that ASP.NET MVC 5 was where we had both System.Web.Mvc and Microsoft.Web.Mvc namespaces – which is confusing. Fortunately this was short-lived and irrelevant today.
• ASP.NET MVC 6 was never released. It was rolled-into ASP.NET 5 which then became ASP.NET Core. Its goal was to combine ASP.NET MVC and ASP.NET Web API into a single platform.
• ASP.NET Core was launched in 2016 as an improved design of ASP.NET MVC 5 but without the dependencies on System.Web.dll or any (now-legacy) IIS dependencies (e.g. the old request pipeline, IHttpHandler and IHttpModule no longer exist).
  • Note that ASP.NET Core‘s class library’s root namespace is now Microsoft.AspNet and not System.Web. This was a source of confusion for me. This means that upgrading projects from ASP.NET MVC and ASP.NET WebAPI to ASP.NET Core are non-trivial, despite their fundamental architecture of Controllers and Middleware being the same.
• .NET Core is a new version of the CLR and BCL which is explicitly portable and runs on Windows, Linux and macOS.
  • .NET Core 1.0 and .NET Core 2.0‘s base class libraries were considered anemic compared to the full-fat .NET Framework which is a Windows-exclusive.
  • With .NET Core 3.0 and the open-sourcing of WinForms, WPF and other components of .NET Framework, .NET Core is now primed as a replacement for .NET Framework in Windows for new application development.
• .NET 5 will be the new name of .NET Core after .NET Core 3 is released (.NET 5 is expected around late 2020).
  • .NET 5 has absolutely nothing to do with ASP.NET 5.0 nor ASP.NET MVC 5.
  • There is no .NET Core 4 nor .NET Framework 5.0.

• “ASP.NET Core in .NET” seems to be the final name for the first release of ASP.NET Core specifically targeting .NET 5 and later.

I note that “Core” is Microsoft’s hot branding for the current generation of .NET platforms which work with .NET Core (i.e. they have no Windows dependencies and so are portable). (Just like how Microsoft stuck “Active” onto things in the 1990s if they used COM or ActiveX, e.g. Active Desktop, Active Channels, ActiMates, Active Directory, ActiveSync, etc).

• _{Additionally Entity Framework Core is still lacking a lot of functionality from Entity Framework 6, this is because it’s a rewrite, basically – but it will reach parity eventually.}

Because there are many .NET runtimes and BCLs currently available (.NET Framework, .NET Core, UWP, Xamarin (which uses Mono), Unity and others) Microsoft introduced .NET Standard which is basically a reboot of the Portable Class Library concept: where Visual Studio projects target a known subset of common functionality instead of a specific implementation. _{(I just wish they’d start the version numbering from 4 to match .NET Framework instead of starting at 1.0 because that got me thinking about 2001 all over again.)} – but the important part is that ASP.NET Core 1 and ASP.NET Core 2 target .NET Standard instead of .NET Core – which means that ASP.NET Core runs on top of the .NET Framework on Windows in addition to running on top of .NET Core as well.

_{I note that all previous .NET cross-platform compatibility techniques are now obsolete (including targeting Compact Framework subsets, Portable Class Libraries, and even “Shared Projects” in Visual Studio), as they were meant for other editions of .NET which no-longer exist, such as .NET Compact Framework, XNA, Silverlight, and Windows Phone 7’s subset.}

In May 2019, Microsoft announced “.NET 5”. In short, the .NET Framework is being replaced with .NET Core, and the next version of .NET Core after .NET Core 3.0 will be named .NET 5). This announcement does not concern ASP.NET Core at all, other than the fact that .NET 5 will fully support ASP.NET Core 3.0 applications running on it.

Throughout the Covid-themed summer of 2020 Microsoft finalized .NET 5 and ASP.NET Core. As of October 2020, ASP.NET Core’s NuGet package versions have jumped from 3.1.x to 5.0.0 – so it looks as though the next major release of ASP.NET Core will be versioned 5.x.x which suggests its release may be named “ASP.NET Core 5” though recent Microsoft blog articles suggest the name will be “ASP.NET Core for .NET 5” – though given that this whole thread was created out of the confusion over “ASP.NET MVC 5” vs “ASP.NET 5” I’m sure that naming it either “ASP.NET Core 5” or “ASP.NET Core for .NET 5” will absolutely not be confusing at all, no sir-ee-bob!

In summary:

• ASP.NET MVC 5:
  • ASP.NET MVC 5 was a short-lived successor to ASP.NET MVC 4.
  • It was released alongside ASP.NET Web API 2 in 2014.
  • It actually ran on top of ASP.NET 4 (i.e. .NET 4.x version of System.Web.dll). Note that the entire ASP.NET MVC library is now obsolete.
• ASP.NET 5 was EOL’d and rebranded as ASP.NET Core and it includes the functionality of “ASP.NET MVC 5” built-in.
• ASP.NET Core 1 and ASP.NET Core 2 can run on either .NET Core (cross-platform) or .NET Framework (Windows) because it targets .NET Standard.
• ASP.NET Core 3 now only runs on .NET Core 3.0.
• ASP.NET Core 4 does not exist and never has.
• ASP.NET Core 5 exists (as of August 2020) however its official name seems to be “ASP.NET Core for .NET 5” and it only runs on .NET 5.

All of them (in chronological order):

• ASP.NET 1 – 2001. Included WebForms. Ran on .NET Framework 1.0 and 1.1. System.Web.dll.
• ASP.NET 2.0 – 2005. Included WebForms. Ran on .NET Framework 2.0. System.Web.dll.
• ASP.NET MVC 1 and ASP.NET MVC 2 – 2008-2009. Ran on top of ASP.NET 2.0. System.Web.Mvc.dll.
• ASP.NET 4.0 – 2010. Included WebForms. Ran on .NET Framework 4.0. There was no ASP.NET 3.0. System.Web.dll.
• ASP.NET MVC 3 and ASP.NET MVC 4 – 2010-2013. Ran on top of ASP.NET 4.0. System.Web.Mvc.dll.
• ASP.NET Web Api 1 – 2012. Ran on top of ASP.NET 4.0. System.Web.Http.dll.
• ASP.NET MVC 5 – 2013. Just another update to ASP.NET MVC. Ran on top of ASP.NET 4.0 but could also run independently without System.Web.dll under OWIN.
• ASP.NET Web API 2 – 2013. Sibling to ASP.NET MVC 5. Could also run without ASP.NET 4.0 under OWIN.
• ASP.NET MVC 6 – 2014-2015. Aborted after reaching Release Candidate status and rebooted as ASP.NET Core MVC 1.0 in 2016 which is the MVC and Web API component of ASP.NET Core 1.
• ASP.NET 5 – 2014. Major reboot of ASP.NET described here. The main changes included merging MVC, Web Pages and Web API – and the removal of WebForms. ASP.NET 5 reached Release Candidate status but was then rebranded as ASP.NET Core. There has never been an ASP.NET 6.
• ASP.NET Core 1 – 2016. Runs on either .NET Framework 4.5 or .NET Core 1.0.
• .NET Core – 2016. Portable and minimal .NET runtime and class library.
• .NET Standard – 2017. A way for programs to target a common set of functionality that will be present in all .NET implementations (.NET Framework 4.5 and later, .NET Core 2.0 and later, Xamarin, etc).
• ASP.NET Core 2 – 2017-2018: where we are today. Runs on either .NET Framework 4.6.1 or .NET Core 2.0. (As of late 2018 there is now ASP.NET Core 2.1).
• ASP.NET Core 3 – In late October 2018 Microsoft announced ASP.NET Core 3.0 will now only run on the upcoming .NET Core 3.0 (so it will no-longer run on the .NET Framework 4.7.x). This is a controversial move because it means there is now no upgrade path from ASP.NET Core 2.x to ASP.NET Core 3.x for applications that run on the .NET Framework 4.7.x because of dependencies that don’t support .NET Core yet, which means there likely won’t be a .NET Standard 3.0.
• .NET 5 – May 2019: Microsoft announced the next version of .NET Core after .NET Core 3.1 would not be called “.NET Core 4.0” nor “.NET Core 5.0” but would jump to version number 5.0 and ditch the “Core” branding, hence “.NET 5.0“.
• ASP.NET Core 3.0 on .NET 5 – In the May 2019 announcement of .NET 5, Microsoft also stated that ASP.NET Core 3.0 applications will run on .NET 5. It is currently unclear if ASP.NET Core 2.x applications will be able to run unmodified on .NET 5.
• ASP.NET Core vNext on .NET 5 – August 2020: With the impending release of .NET 5, Microsoft has bumped the version numbers of the NuGet packages for ASP.NET Core to 5.0.0. Microsoft has not yet formally/officially referred to this new release of ASP.NET Core as “ASP.NET Core 5”, but I see it as a foregone conclusion.
• ASP.NET Core for .NET 5 A more recent blog article from September 2020 suggests that Microsoft is referring to the next version of ASP.NET Core as “ASP.NET Core for .NET 5” (yes, that’s a mouthful – at least it isn’t as ambiguous as earlier names).
  • Update (September 2021): It seems that the official name is now ASP.NET Core in .NET 5, and for .NET 6 it seems they’ve gone with ASP.NET Core in .NET 6 as well.

Timeline and block diagram

(I got carried away making this…)