This paper proposes and studies for the first time the problem of combinatorial multi-armed bandits with linear long-term constraints. Our model generalizes and unifies several prominent lines of work, including bandits with fairness constraints, bandits with knapsacks (BwK), etc. We propose an upper-confidence bound LP-style algorithm for this problem, called UCB-LP, and prove that it achieves a logarithmic problem-dependent regret bound and zero constraint violations in expectation. In the special case of fairness constraints, we further provide a sharper constant regret bound for UCB-LP. Our regret bounds outperform the existing literature on BwK and bandits with fairness constraints simultaneously. We also develop another low-complexity version of UCB-LP and show that it yields $\tilde{O}(\sqrt{T})$ problem-independent regret and zero constraint violations with high-probability. Finally, we conduct numerical experiments to validate our theoretical results.