Microsoft's official position:

Relationship to COM

One of the primary goals of the .NET Framework is to make COM development easier. One of the hardest things about COM development is simply dealing with the COM infrastructure. Consequently, to make COM development easier, the .NET Framework automates virtually all of what developers currently think of as “COM,” including reference-counting, interface description, and registration.

It’s important to note that this doesn’t mean that .NET Framework components aren’t COM components. In fact, a COM developer using Visual Studio 6.0 could call a .NET Framework component and, to the developer, it would look like a COM component, complete with iUnknown data. Conversely, a .NET Framework developer using Visual Studio.NET would see a COM component as a .NET Framework component.

There is a caveat to this relationship: COM developers must manually do many of the things that .NET Framework developers can rely on the runtime to automate for them. For example, the security of a COM component must be written manually, and its memory can’t be automatically managed, and, to install a COM component, entries must be placed in the Windows registry. For .NET Framework components, the runtime automates these features. Components are self-describing, for example, and can therefore be installed without registering them in the Windows registry.

Relationship to COM+

COM+ is the name of COM when you combine it with Microsoft Transaction Server (MTS), and Distributed COM (DCOM). COM+ provides a set of middle-tier-oriented services. In particular, COM+ provides process management and database and object connection pooling. In future versions, it will also provide stronger process isolation designed for application service providers—a feature called partitioning.

The COM+ services are primarily oriented toward middle-tier application development and focus on providing reliability and scalability for large-scale, distributed applications. These services are complementary to the programming services provided by the .NET Framework; the .NET Framework classes provide direct access to them.