The article provides a critical review of the M_{n+1}AX_{n} phases, also known as MAX phases, from a materials science perspective. These phases are a class of hexagonal-structure ternary carbides and nitrides composed of a transition metal (M), an A-group element (A), and a non-metal (X, typically C or N). The most well-known MAX phases include Ti₂AlC, Ti₃SiC₂, and Ti₄AlN₃. Over 60 MAX phases have been identified, with many discovered in the last five years. These phases are unique due to their combination of metallic and ceramic properties, including high thermal and electrical conductivity, resistance to oxidation and corrosion, and mechanical strength. Their nanolaminated structure, with alternating M_{n+1}X_{n} and A layers, contributes to these properties. The research on MAX phases has been accelerated by the development of thin-film processing methods, such as magnetron sputtering and arc deposition, which allow for the synthesis of single-crystal materials. However, the decomposition of MAX phases at high temperatures is lower than that of their bulk counterparts. Recent advances in low-temperature synthesis have enabled the deposition of MAX phases onto technologically important substrates. For example, V₂GeC and Cr₂AlC can be deposited at around 450°C. The article also discusses the prospects for low-temperature synthesis, the relationship between thin-film and bulk synthesis, and the use of first-principles calculations to predict new MAX phases. It highlights the importance of understanding the physical properties of MAX phases, including the effects of anisotropy, impurities, and vacancies on their behavior. The article concludes with a discussion of future research directions, including the exploration of unknown regions in phase diagrams to discover new equilibrium and metastable phases, as well as the challenges in understanding the physical properties of MAX phases.The article provides a critical review of the M_{n+1}AX_{n} phases, also known as MAX phases, from a materials science perspective. These phases are a class of hexagonal-structure ternary carbides and nitrides composed of a transition metal (M), an A-group element (A), and a non-metal (X, typically C or N). The most well-known MAX phases include Ti₂AlC, Ti₃SiC₂, and Ti₄AlN₃. Over 60 MAX phases have been identified, with many discovered in the last five years. These phases are unique due to their combination of metallic and ceramic properties, including high thermal and electrical conductivity, resistance to oxidation and corrosion, and mechanical strength. Their nanolaminated structure, with alternating M_{n+1}X_{n} and A layers, contributes to these properties. The research on MAX phases has been accelerated by the development of thin-film processing methods, such as magnetron sputtering and arc deposition, which allow for the synthesis of single-crystal materials. However, the decomposition of MAX phases at high temperatures is lower than that of their bulk counterparts. Recent advances in low-temperature synthesis have enabled the deposition of MAX phases onto technologically important substrates. For example, V₂GeC and Cr₂AlC can be deposited at around 450°C. The article also discusses the prospects for low-temperature synthesis, the relationship between thin-film and bulk synthesis, and the use of first-principles calculations to predict new MAX phases. It highlights the importance of understanding the physical properties of MAX phases, including the effects of anisotropy, impurities, and vacancies on their behavior. The article concludes with a discussion of future research directions, including the exploration of unknown regions in phase diagrams to discover new equilibrium and metastable phases, as well as the challenges in understanding the physical properties of MAX phases.