Terahertz (THz) communications (THzCom) has experienced a meteoric rise of interest, due to its benefits for ultra-high data rate transmission in the sixth generation (6G) and beyond era. Despite so, the research on exploring the potential of THzCom for other performance targets anticipated by 6G, including massive connectivity and security, is still in its infancy. In this article, we start with briefly describing the unique peculiarities of THz channels, and then discuss theoretical frameworks to facilitate the analysis and design of THz transmission for achieving massive connectivity and security. Then we discuss promising spectrum management strategies, including the exploration of multiple THz transmission windows and frequency reuse with multiplexing and signal processing, to substantially increase the number of supported users and identify to-be-tackled challenges. We further present important research directions based on the principles of physical layer security, such as new spectrum allocation policies and beamforming algorithms, to fight against eavesdropping in THzCom systems, ushering in secure THzCom systems.