Model Groups are really powerful to duplicate repeating elements (or element groups) in a model. However, their limitations restrain users to use these to the full extent. I made a summary of the most frequently occurring issues.
Limitation 1: Assign parameters to Detail Groups
It’s not possible to add Shared Parameters to Detail Groups or create a Detail Group Schedule.
Limitation 2: Nested Model Groups
Create a Model Group (Group 1). Create a second Model Group (Group 2). Add Group 2 to Group 1 using Add to Group. Change the Group Type of Group 1 to Group 2. This error will appear:

Of course the proposed workaround is possible, but not desirable at all.
Limitation 3: System Type of exported Groups
- Draw some Ducts or Pipes and assign a System Type
- Create a Model Group containing these Ducts or Pipes
- Export this Model Group to .rvt and open this file.
- Result: the System Type of the exported elements is set to Undefined:

Same issue appears when the Model Group is converted to a Link:

Limitation 4: Moving Groups with hosted elements on multiple faces
After elements which hosted on elements in a linked .rvt are grouped (on multiple faces in several directions like Floors and Walls), they cannot be moved or copied propertly without ungrouping all the elements or creating new Group types (“Fix Groups…”):

[API] Limitation 5: Access to Edit Group mode for external add-ins
We’d like to modify elements in Edit Group mode. Therefore lots of functionality will become available in Edit Group mode, which can then be applied to all Group instances easily. Currently the whole Add-Ins Panel is disabled in Edit Group mode:

[API] Limitation 6: Set Origin Point
We’d like access to the Origin Point of Model Groups and Detail Groups.
[API] Limitation 7: Add element(s) to existing Model Group
Adding an element to an existing Model Group is not possible. The current workaround is to Ungroup the Model Group, add the element and create a new Model Group, which has a lot of side effects when hosted families are used.