The ability of network users to influence other users’ behavior has deep implications for fields such as marketing, epidemiology, misinformation control, and cybersecurity. While traditional models have extensively studied the cascading effects of influence without considering the speed of propagation, this work highlights methodologies that account for this aspect, reviewing the recent advancements in the study of rapid influence (i.e., influence propagation limited to a fixed number of hops or within a minimal time frame from the initial seed set). We provide a comprehensive review of the different variants of this problem studied in the literature, discussing both the theoretical aspects and practical implications, as well as the proposed solution approaches.