Protein structure prediction (PSP) is a grand challenge in bioinformatics, drug discovery, and related fields. PSP is computationally challenging because of an astronomically large conformational space to be searched and an unknown very complex energy function to be minimised. To obtain a given protein’s structure, template-based PSP approaches adopt a similar protein’s known structure, while template-free PSP approaches work when no similar protein’s structure is known. Currently, proteins with known structures are greatly outnumbered by proteins with unknown structures. Template-free PSP has obtained significant progress recently via machine learning and search-based optimisation approaches. However, very accurate structures for complex proteins are yet to be achieved at a level suitable for effective drug design. Moreover, ab initio prediction of a protein’s structure only from its amino acid sequence remains unsolved. Furthermore, the number of protein sequences with unknown structures is growing rapidly. Hence, to make further progress in PSP, more sophisticated and advanced artificial intelligence (AI) approaches are needed. However, getting involved in PSP research is difficult for AI researchers because of the lack of a comprehensive understanding of the whole problem, along with the background and the literature of all related sub-problems. Unfortunately, existing PSP review papers cover PSP research at a very high level and only some parts of PSP and only from a particular singular viewpoint. Using a systematic approach, this review paper provides a comprehensive survey of the state-of-the-art template-free PSP research to fill this knowledge gap. Moreover, covering required PSP preliminaries and computational formulations, this paper presents PSP research from AI perspectives, discusses the challenges, provides our commentaries, and outlines future research directions.