Spectrally efficient frequency division multiplexing (SEFDM) is a multi-carrier signal waveform, which achieves higher spectral efficiency, relative to conventional orthogonal frequency division multiplexing (OFDM), by violating the orthogonality of its sub-carriers. This survey provides the history of SEFDM development since its inception in 2003, covering fundamentals and concepts, wireless and optical communications applications, circuit design and experimental testbeds. We focus on work done at UCL and outline work done other universities and research laboratories worldwide. We outline techniques to improve the performance of SEFDM and its practical utility with focus on signal generation, detection and channel estimation.