As the enabling technolgies move up to the mmWave and even to the TeraHertz bands for the next-generation wireless systems, the signal processing of high-bandwidth orthogonal frequency division multiplexing (OFDM) becomes increasingly power-thirsty, owing to the following OFDM deficiencies: (1) the high peak-to-average power ratio (PAPR); (2) the bandwidth efficiency loss due to the cyclic prefix (CP) overhead; (3) the sensitivity to carrier frequency offset; (4) the complex out-ofband (OOB) filtering. Over the past six decades, a variety of waveforms have been developed in order to mitigate these deficiencies, which are generally achieved at the cost of compromising some of OFDM's beneficial properties, such as its subcarrier (SC) orthogonality, its high throughput and its straighforward adoption to multiple-input multiple-output (MIMO) systems. Against this background, we propose a new waveform termed as multi-band discrete Fourier transform spread-OFDM with index modulation (MB-DFT-S-OFDM-IM), where the component multi-carrier techniques are conceived to constructively function together in order to mitigate the OFDM deficiencies without compromising the beneficial OFDM properties. More explicitly, first of all, the PAPR is reduced by the DFT-precoding. Secondly, thanks to the IM design, MB-DFT-S-OFDM-IM is capable of achieving a high throughput that is strictly equal to or higher than the OFDM throughput. Thirdly, MB-DFT-S-OFDM-IM achieves a beneficial frequency diversity gain, which leads to a higher tolerance to carrier frequency offset. Fourthly, the OOB filters are placed in each sub-band before DFT, so that the SC orthogonality remains intact, which is unique to the proposed MB-DFT-S-OFDM-IM structure. Last but not least, we extend the proposed MB-DFT-S-OFDM-IM to support a variety