MME can be assumed as the main control node of the LTE network and It has the following functionality.
Network Access Control: MME manages authentication and authorization for the UE. It also facilitates UE access to the network to gain IP connectivity.
Radio Resource Management: MME works with the HSS and the RAN to decide the appropriate radio resource management strategy (RRM) that can be UE-specific.
Mobility Management: One of the most complex functions MME performs. Providing seamless inter-working has multiple use cases such as Inter-eNB and Inter-RAT, among others. The use cases become more complex depending on a change in MME, S-GW, P-GW or inter-working across other wireless networks.
Roaming Management: MME supports outbound and inbound roaming subscribers from other LTE/EPC systems and legacy networks.
UE Reach-ability: MME manages communication with the UE and HSS to provide UE reach-ability and activity-related information.
Tracking Area Management: Allocates and reallocates a tracking area identity list to the UE.
Lawful Intercept: Since MME manages the control plane of the network, MME can provide the whereabouts of a UE to a law enforcement monitoring facility.
Load Balancing Between S-GWs: Directs UEs entering an S-GW pool area to an appropriate S-GW. This achieves load balancing between S-GWs.
Following image should help...