Conceptual design is the foundational stage of a design process that translates ill-defined design problems into low-fidelity design concepts and prototypes through design search, creation, and integration. In this stage, product shape design is one of the most paramount aspects. When applying deep learning-based methods to product shape design, two major challenges exist: (1) design data exhibit in multiple modalities and (2) an increasing demand for creativity. With recent advances in deep learning of cross-modal tasks (DLCMTs), which can transfer one design modality to another, we see opportunities to develop artificial intelligence (AI) to assist the design of product shapes in a new paradigm. In this paper, we conduct a systematic review of the retrieval, generation, and manipulation methods for DLCMT that involve three cross-modal types: text-to-3D shape, text-to-sketch, and sketch-to-3D shape. The review identifies 50 articles from a pool of 1341 papers in the fields of computer graphics, computer vision, and engineering design. We review (1) state-of-the-art DLCMT methods that can be applied to product shape design and (2) identify the key challenges, such as lack of consideration of engineering performance in the early design phase that need to be addressed when applying DLCMT methods. In the end, we discuss the potential solutions to these challenges and propose a list of research questions that point to future directions of data-driven conceptual design.