This paper investigates the impact of deploying the fluid antenna system (FAS) on the performance of covert communications. In particular, we focus on a scenario where a transmitter seeks to covertly communicate with a receiver, while a warden attempts to detect the transmission. Both the receiver and the warden are assumed to utilize planar FAS. We derive compact analytical expressions for the covertness outage probability (COP), defined as the complement of the sum of false alarm (FA) and missed detection (MD) probabilities. By determining the optimal detection threshold that maximizes the COP, we characterize the success probability for the legitimate transmission, highlighting the trade-off between covertness and transmission success. Our numerical results confirm that while deploying FAS at the warden enhances its detection ability compared to fixed-position antennas (FPAs), equipping the receiver with FAS rather than FPAs significantly improves reception quality, leading to more reliable transmission.