In this paper, we provide a common generalization to the well-known Erdős-Ko-Rado Theorem, Frankl-Wilson Theorem, Alon-Babai-Suzuki Theorem, and Snevily Theorem on set systems with $\mathcal{L}$-intersections. As a consequence, we derive a result which strengthens substantially the well-known theorem on set systems with $k$-wise $\mathcal{L}$-intersections by F$\ddot{u}$redi and Sudakov [J. Combin. Theory, Ser. A (2004) 105: 143-159]. We will also derive similar results on $\mathcal{L}$-intersecting families of subspaces of an $n$-dimensional vector space over a finite field $\mathbb{F}_{q}$, where $q$ is a prime power.