The rapid progress of artificial intelligence (AI) has led to the emergence of a highly promising field known as neuro-symbolic (NeSy) computing. This approach combines the strengths of neural networks, which excel at data-driven learning, with the reasoning capabilities of symbolic AI. Neuro-symbolic models have the potential to overcome the limitations of each approach individually, resulting in interpretable and explainable AI systems that can reason over complex knowledge bases, learn from limited and/or noisy data, and be generalizable. However, the exploration of NeSy AI from a system perspective remains limited. This brief provides an in-depth analysis of the state-of-the-art hardware-software co-design techniques for NeSy AI and discusses the associated challenges in improving system efficiency for heterogeneous computing. By examining the intersection of NeSy computing and system design, we aim to bridge the gap and foster advancements in this domain.