The novel concept of non-orthogonal multiple access (NOMA) aided joint radar and multicast-unicast communication (Rad-MU-Com) is investigated. Employing the same spectrum resource, a multi-input-multi-output (MIMO) dual-functional radar-communication (DFRC) base station detects the radarcentric users (R-user), while transmitting mixed multicast-unicast messages both to the R-user and to the communication-centric user (C-user). In particular, the multicast information is intended for both the R-and C-users, whereas the unicast information is only intended for the C-user. More explicitly, NOMA is employed to facilitate this double spectrum sharing, where the multicast and unicast signals are superimposed in the power domain and the superimposed communication signals are also exploited as radar probing waveforms. First, a beamformer-based NOMA-aided joint Rad-MU-Com framework is proposed for the system having a single R-user and a single C-user. Based on this framework, the unicast rate maximization problem is formulated by optimizing the beamformers employed, while satisfying the rate requirement of multicast and the predefined accuracy of the radar beam pattern. The resultant non-convex optimization problem is solved by a penalty-based iterative algorithm to find a high-quality nearoptimal solution. Next, the system is extended to the scenario of multiple pairs of R-and C-users, where a cluster-based NOMA-aided joint Rad-MU-Com framework is proposed. A joint beamformer design and power allocation optimization problem is formulated for the maximization of the sum of the unicast rate at each C-user, subject to the constraints on both the minimum multicast rate for each R&C pair and on accuracy of the radar beam pattern for detecting multiple R-users. The resultant joint optimization problem is efficiently solved by another penalty-based iterative algorithm developed. Finally, our numerical results reveal that significant performance gains can be achieved by the proposed schemes over the benchmark schemes employing conventional transmission strategies.