We present a novel bandwidth (BW) efficient orthogonal frequency division multiplexing (OFDM) scheme with adaptive zero-padding (AZP-OFDM) for wireless transmission. Redundancy issues in OFDM based on cyclic prefix (CP), zero-padding (ZP), as well as no guard interval (NGI) systems are analyzed. A novel system design criterion based on the channel matrix condition is studied and applied to the design of an AZP-OFDM system. Simulation results have shown that the proposed AZP-OFDM offers performance similar to that of CP-OFDM, complexity similar to that of ZP-OFDM, with BW efficiency higher than that of both CP- and ZP-OFDM in channels with small to moderate delay spread. In channels with large delay spread, AZP scheme adaptively maintains high performance at the expense of BW efficiency. Essentially, AZP-OFDM offers a more flexible tradeoff between symbol recovery, BW efficiency, and complexity.