Traditionally, speech emotion recognition (SER) research has relied on manually handcrafted acoustic features using feature engineering. However, the design of handcrafted features for complex SER tasks requires significant manual effort, which impedes generalisability and slows the pace of innovation. This has motivated the adoption of representation learning techniques that can automatically learn an intermediate representation of the input signal without any manual feature engineering. Representation learning has led to improved SER performance and enabled rapid innovation. Its effectiveness has further increased with advances in deep learning (DL), which has facilitated deep representation learning where hierarchical representations are automatically learned in a data-driven manner. This paper presents the first comprehensive survey on the important topic of deep representation learning for SER. We highlight various techniques, related challenges and identify important future areas of research. Our survey bridges the gap in the literature since existing surveys either focus on SER with hand-engineered features or representation learning in the general setting without focusing on SER.