Nonlocality has become increasingly noticeable in nature. New mathematical tools and computational algorithms are in need to address the challenges involved in the modeling and simulation of nonlocal interactions. In this talk, we focus on nonlocal models with a finite horizon of interactions and their roles in various applications. We reveal some practical risks due to inappropriately designed nonlocal models and discretizations. We show how systematic mathematical development can help making nonlocal modeling and simulations more effective and robust for problems in areas like solid and fluid mechanics, and autonomous vehicle traffic flows.