The electricity sector plays a key role in achieving zero emissions targets. The required transition will lead to substantial changes in the supply, demand, and distribution of electricity, as well as in stakeholder roles. Future market designs may change substantially to accommodate these changes, address challenges, and take advantage of new opportunities. This paper reviews the characteristics of future carbon-neutral electricity systems and electricity market design options. To provide a guiding framework for the literature review, we transfer the complexity of electricity systems into a three-layer structure: Firstly, we analyze papers that rely on techno-economic modeling of the physical electricity system. As a case study, we analyze various studies focusing on a decarbonized European electricity system in 2050. Secondly, we review papers that investigate the economic behavior and effects of self-interest-seeking stakeholders such as producers, network operators, and consumers. Finally, we review papers focusing on policy and market design questions that guide policymakers in achieving a target physical asset combination while considering the behavior of stakeholders. We highlight common trends and disagreements in the literature, review the main drivers of future markets, and finally provide a mapping between those drivers, challenges, and opportunities. The review concludes that the most promising next step toward a fully comprehensive assessment approach is to combine existing approaches across topical and disciplinary boundaries.