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In this paper, a novel non-orthogonal multiple access (NOMA)-based device-to-device (D2D) communications framework is proposed. A major novelty of the proposed framework is that it introduces the new concept of "D2D group" which utilizes NOMA transmission, enabling one D2D transmitter to communicate with multiple D2D receivers simultaneously. Based on the considered framework, a resource allocation optimization problem is formulated, where multiple D2D groups are allowed to reuse the same subchannel. The objective of this work is to maximize the system sum rate by satisfying the signalto-interference-plus-noise (SINR) constraints of both D2D and traditional cellular users. Note that the formulated problem is non-deterministic polynomial-time (NP) hard in nature, thus a novel resource allocation algorithm based on the many-to-one twosided matching theory is proposed for obtaining a suboptimal solution. It is proved that the proposed algorithm converges to a stable state within limited number of iterations. Numerical results illustrate that: i) the proposed algorithm is an effective approach for obtaining near-optimal performance with acceptable complexity; and ii) the proposed NOMA-based D2D framework is capable of achieving promising gains over traditional orthogonal multiple access (OMA)-based D2D framework.