Multi-carrier systems suffer from several impairments, and communication system engineers use powerful signal processing tools to combat these impairments and keep up with the capacity/rate demands. Compressed sensing (CS) is one such tool that allows recovering any sparse signal, requiring only a few measurements in a domain that is incoherent with the domain of sparsity. Almost all signals of interest have some degree of sparsity, and in this work we utilize the sparsity of impairments in orthogonal frequency division multiplexing (OFDM) and its variants (i.e., orthogonal frequency division multiplexing access (OFDMA) and single-carrier frequency-division multiple access (SC-FDMA)) to combat them using CS. We start with the problem of peak-to-average power ratio (PAPR) reduction in OFDM. OFDM signals suffer from high PAPR and clipping is the simplest PAPR reduction scheme. However, clipping introduces inband distortions that result in compromised performance and hence needs to be mitigated at the receiver. Due to the high PAPR nature of the OFDM signal, only a few instances are clipped, these clipping distortions can be recovered at the receiver by employing CS. We then extend the proposed clipping recovery scheme to an interleaved OFDMA system. Interleaved OFDMA presents a special structure that result in only self-inflicted clipping distortions. In this work, we prove that distortions do not spread over multiple users (while utilizing interleaved carrier assignment in OFDMA) and construct a CS system that recovers the clipping distortions on each user. Finally, we address the problem of narrowband interference (NBI) in SC-FDMA. Unlike OFDM and OFDMA systems, SC-FDMA does not suffer from high PAPR, but (as the data is encoded in time domain) is seriously vulnerable to information loss owing to NBI. Utilizing the sparse nature of NBI (in frequency domain) we combat its effect on SC-FDMA system by CS recovery.