I've used a Moodle-based LMS at two different schools now and am overall happy with the functionality and features it offers me as a user. Managing course content is for the most part intuitive and easy. User experience and options are dependent on how Moodle is configured by the institution's IT or learning technology team/department, and the implementation I've seen has varied. Some deficiencies I noticed at one school were not present at the other and I discovered it was a result of how Moodle was managed on the back end. For example, one college did not enable the very simple text formatting in input windows, and the other college had not enabled the attendance logging features. So, if there are features you are looking for, I recommend reading the documentation on the Moodle site to find out if they are available.
Strengths: online help, customizability, ease of use (creating and managing content), assignment and grade management, transferring/importing files and learning modules across semesters, assignment/submission management, adaptability, interactivity.
As mentioned, Moodle is a solid LMS solution in my experience as a user, but I cannot speak to the back end enterprise software management side. Two issues I encounter that have more to do with efficiency than "problems," etc.
* Moving Course Content within a Page: The issue I find most problematic as a user is navigating long Moodle pages for courses with heavy online content for the purposes of moving content modules between weeks. Pages can become very long and while the drag and drop feature is great, it is difficult to do when attempting to drag a module from let's say week 3 to week 9 - the scrolling down the page is sluggish, stutters, and inefficient.
* Menus/Access to Editing Modules: The process for editing week sections, labels, assignments, etc., is less than convenient and intuitive. It seems there would be an "edit" icon next to each entry on a Moodle page to allow users to click and immediately go to the edit interface - often it is an icon that looks like a pencil and paper. Instead, users must click a settings icon that displays a pop-up menu with options that includes editing.
* Search Capabilities: There is currently not much in the way of search capabilities within Moodle pages and user-created content.
* Intuitive/Ease of Use: Moodle is less than intuitive in some areas, and while Moodle training (as with any software package/solution) requires some level of training, there are some features that could be more intuitive. Your mileage varies when it comes to training large groups of users and so making some features more intuitive would be helpful, especially for users who are not as computer/web/tech savvy as others.
Refer to my other responses regarding recommendations.
In my experience, Moodle is strong in the LMS categories I have observed: content authoring and management, learning management, course library functions, user management, and online testing and quizzes.