This paper investigates the application of fast-convolution (FC) filtering schemes for flexible and effective waveform generation and processing in the fifth generation (5G) systems. FC-based filtering is presented as a generic multimode waveform processing engine while, following the progress of 5G new radio standardization in the Third-Generation Partnership Project, the main focus is on efficient generation and processing of subband-filtered cyclic prefix orthogonal frequency-division multiplexing (CP-OFDM) signals. First, a matrix model for analyzing FC filter processing responses is presented and used for designing optimized multiplexing of filtered groups of CP-OFDM physical resource blocks (PRBs) in a spectrally well-localized manner, i.e., with narrow guardbands. Subband filtering is able to suppress interference leakage between adjacent subbands, thus supporting independent waveform parametrization and different numerologies for different groups of PRBs, as well as asynchronous multiuser operation in uplink. These are central ingredients in the 5G waveform developments, particularly at sub-6-GHz bands. The FC filter optimization criterion is passband error vector magnitude minimization subject to a given subband band-limitation constraint. Optimized designs with different guardband widths, PRB group sizes, and essential design parameters are compared in terms of interference levels and implementation complexity. Finally, extensive coded 5G radio link simulation results are presented to compare the proposed approach with other subband-filtered CP-OFDM schemes and time-domain windowing methods, considering cases with different numerologies or asynchronous transmissions in adjacent subbands. Also the feasibility of using independent transmitter and receiver processing for CP-OFDM spectrum control is demonstrated