In the realm of next-generation wireless systems (beyond 5G/6G), the vision is clear: to support a broad range of services and applications. This includes ensuring reliable communications in environments marked by high mobility, such as high-speed railway systems and various vehicular communications. Despite the deployment of various multicarrier techniques like orthogonal frequency division multiplexing (OFDM) and single-carrier frequency division multiple access (SC-FDMA) in standardized communication systems, the challenge persists. These techniques, while effective in time-invariant frequency selective channels, face performance degradation in high mobility scenarios due to the destruction of orthogonality among subcarriers caused by significant Doppler frequency shifts. Addressing this, the search for new, robust modulation techniques is paramount. It stands as a key area of investigation aiming to resolve the reliable communications issue for next-generation wireless networks within doubly-selective wireless channels. In this thesis, a novel solution, affine frequency division multiplexing (AFDM), is proposed. This new chirp-based multicarrier waveform is based on the discrete affine Fourier transform (DAFT), a variant of the discrete Fourier transform characterized with two parameters that can be adapted to better cope with doubly dispersive channels. This thesis provides a comprehensive investigation into the principles of AFDM within high mobility communications. It provides insight into the explicit input-output relation in the DAFT domain, unveiling the consequential impact of AFDM parameters. The manuscript details the precise setting of DAFT parameters, ensuring a full delay-Doppler representation of the channel. Through analytical demonstrations, it asserts that AFDM optimally achieves the diversity order in doubly dispersive channels due to its full delay-Doppler representation. The thesis also proposes two low-complexity detection algorithms for AFDM, taking advantage of its inherent channel sparsity. The first is a low complexity MMSE detector based on LDL factorization. The second is a low complexity iterative decision feedback equalizer (DFE) based on weighted maximal ratio combining (MRC) of the channel impaired input symbols received from different paths. Additionally, the thesis presents an embedded channel estimation strategy for AFDM systems, leveraging AFDM's ability to achieve full delay-Doppler representation of the channel. In this approach, an AFDM frame contains a pilot symbol and data symbols, with zero-padded symbols employed as guard intervals to prevent interference. A practical channel estimation algorithm based on an approximate maximum likelihood (ML) approach and compatible with this pilot scheme is also provided. The thesis concludes by delving into the expanded applications of AFDM, specifically in integrated sensing and communication (ISAC) and extremely high frequency (EHF) band communications. It is demonstrated that to identify all delay and Doppler components linked with the propagation medium, one can use either the full AFDM signal or only its pilot part consisting of one DAFT domain symbol and its guard interval. Furthermore, the chirp nature of AFDM allows for unique and simple self-interference cancellation with a single pilot, eliminating the need for costly full-duplex methods. The thesis also highlights AFDM's efficient performance in high-frequency bands (with or without mobility), where the maximal spreading of its signal in time and frequency ensures a coverage gain. Unlike other waveforms, AFDM not only provides maximal time-frequency spreading but also ensures robust and efficient detection, characterized by one-tap equalization and resilience to carrier frequency offset (CFO) and phase noise.