Current generation mobile communications require high-quality services. Adopting multiple access (MA) and multi-carrier waveforms potentially enhances the quality of services offered to end-users. However, in the majority of literature, the integration of multi-carrier and multiple-access approaches have not been extensively examined. One possible solution is to review multiple access and multi-carrier waveforms simultaneously to create a favorable foundation for the integration of these schemes. Thus, we consider a comprehensive review of multiple-access systems and multi-carrier waveforms jointly from 1{st} to 5{th} -generation (1G-5G) cellular networks. Initially, we present orthogonal MA (OMA) schemes called: frequency division multiple access (FDMA), time division multiple access (TDMA), code division multiple access (CDMA), and orthogonal frequency division multiple access (OFDMA) that have been utilized in 1G, 2G, 3G, and 4G, respectively. In addition, 5G wireless non-orthogonal multiple access (NOMA) techniques such as power domain NOMA (PD-NOMA), code domain NOMA (CD-NOMA), and other NOMA multiplexing methods are addressed in detail. On the other hand, we glanced at 5G cellular multi-carrier waveforms such as filter bank multi-carrier (FBMC), universal filtered multi-carrier (UFMC), generalized frequency division multiplexing (GFDM), and filtered orthogonal frequency division multiplexing (f-OFDM) waveforms. The assessment and comparison between different OMA, NOMA, and multi-carrier waveforms are carried out with the parameters: modulation schemes, bit error rate (BER), signal to noise ration (SNR), sum rate, peak-to-average power ratio (PAPR), latency, out of band emission (OOBE), and complexity. The analytical formulas of OMA, NOMA, and multi-carrier waveform schemes are also derived and verified using simulation data. Each multiple access strategy's merits, shortcomings, applications, and factors influencing its performance are also addressed. Eventually, possible recommendations for the integration of multiple-access and modulation technologies for next-generation mobile networks are also included.
