Terahertz communications is one of the promising technologies for the future mobile communication networks. Due to the ever increasing need for high-speed data transfer, higher bandwidths are required. As a solution, higher frequencies in the spectrum may be considered for future networks. In this paper, Terahertz band is selected since the latest technological advances make it possible to deploy mobile networks in this high-frequency band. To support higher bandwidth with an advanced multiple access method, non-orthogonal multiple access (NOMA) is implemented in this paper. Different from orthogonal multiple access (OMA), NOMA makes it possible for users to share the same frequency and time resources simultaneously, therefore increases the spectral efficiency. To measure the efficiency of the implemented methods, computer simulations are used and results are obtained for different scenarios. In the simulation results, it is shown that NOMA provides higher average data rates for these scenarios where users are located on variety of distances far from the Terahertz access point. As a result, this work shows NOMA for Terahertz communication networks is a promising method for future communication networks.