The design of flexible road pavements is a complex process as a result of the multiple variables that influence and interact in the models that allow the design of each layer. In recent years, a particular interest has been raised to ensure that climate is considered in pavement design due to temperature and precipitation that influence the deterioration of pavements, impacting their service life. This paper presents a critical review of flexible pavement design methods, from the first ones based on experience, such as empirical methods, to the most recent ones on mechanical–empirical methodologies, where, based on different principles, they determine the thicknesses of the layers that integrate the structure of a pavement to identify how these methods have included climate variables within their methodology. Through this review, it was identified that temperature is incorporated in the dynamic modulus of the asphalt mix, and precipitation and moisture are incorporated through the resilient modulus in the granular layers (base, subbase, and foundation soil or subgrade courses). As a result, it was identified that the most holistic way of integrating climate is through the Enhanced Integrated Climatic Model (EICM) from the Mechanistic–Empirical Pavement Design Guide (MEPDG). In many cases, climate is incorporated through parameters whose behavior is associated with temperature and precipitation but does not use the data of these climate variables directly from the project site. The practical incorporation of climate into design methods allows an increase in the certainty of results, ensuring additional climate-resilient pavement structures and increasing their durability and sustainability during their service life.