We present Affine Frequency Division Multiplexing (AFDM), a new chirp-based multicarrier transceiver scheme for high mobility communications in next-generation wireless systems. AFDM is based on discrete affine Fourier transform (DAFT), a generalization of discrete Fourier transform characterized with two parameters that can be adapted to better cope with doubly dispersive channels. Based on the derived input-output relation, the DAFT parameters underlying AFDM are set in such a way to avoid that time domain channel paths with distinct delays or Doppler frequency shifts overlap in the DAFT domain. The resulting DAFT domain impulse response thus conveys a full delay-Doppler representation of the channel. We show that AFDM can achieve the full diversity of linear time-varying (LTV) channels. Our analytical results are validated through numerical simulations, which evince that AFDM outperforms state-of-the-art multicarrier schemes in terms of bit error rate (BER) in doubly dispersive channels.