Climate change can have economic consequences, affecting the nutritional intake of populations and increasing food insecurity, as it negatively affects diet quality parameters. One way to mitigate these consequences is to change the way we produce and consume our food. A healthy and sustainable diet aims to promote and achieve the physical, mental, and social well-being of the populations at all life stages, while protecting and safeguarding the resources of the planet and preserving biodiversity. Over the past few years, several indexes have been developed to evaluate dietary sustainability, most of them based on the EAT-Lancet reference diet. The present review explains the problems that arise in human nutrition as a result of climate change and presents currently available diet sustainability indexes and their applications and limitations, in an effort to aid researchers and policy actors in identifying aspects that need improvement in the development of relevant indexes. Overall, great heterogeneity exists among the indicators included in the available indexes and their methodology. Furthermore, many indexes do not adequately account for the diets’ environmental impact, whereas others fall short in the economic impact domain, or the ethical aspects of sustainability. The present review reveals that the design of one environmentally friendly diet that is appropriate for all cultures, populations, patients, and geographic locations is a difficult task. For this, the development of sustainable and healthy diet recommendations that are region-specific and culturally specific, and simultaneously encompass all aspects of sustainability, is required.