This article provides a critical review of the M$_{n+1}$AX$_n$ phases, known as MAX phases, from a materials science perspective. MAX phases are a class of hexagonal-structured ternary carbides and nitrides (X) formed by a transition metal (M) and an A-group element. The most well-known MAX phases include Ti$_2$AlC, Ti$_3$SiC$_2$, and Ti$_4$AlN$_3$. These phases exhibit unique combinations of chemical, physical, electrical, and mechanical properties, making them potentially useful in various applications such as high-temperature structural materials, protective coatings, sensors, and low-friction surfaces. 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 by epitaxial growth. However, the surface-initiated decomposition of MAX phases into MX compounds at temperatures of 1000–1100 °C is lower than the decomposition temperatures typically reported for bulk materials. Recent advancements in low-temperature synthesis, such as the deposition of V$_2$GeC and Cr$_2$AlC at ~450 °C, have also been discussed. The article reviews the progress in first-principles calculations for predicting hypothetical MAX phases and their properties, leading to the discovery of new MAX phases like Ti$_4$SiC$_3$, Ta$_4$AlC$_3$, and Ti$_3$SnC$_2$. Future research directions include charting unknown regions in phase diagrams, understanding the effects of anisotropy, impurities, and vacancies on electrical properties, and exploring unexplored properties such as superconductivity, magnetism, and optics. The article also covers the fundamentals of MAX phases, including their crystal structure, phase diagrams, and terminology. It discusses polymorphism, intergrown structures, vacancies, and solid solutions in MAX phases. The thin-film processing methods, such as physical vapor deposition (PVD), chemical vapor deposition (CVD), and solid-state reaction synthesis, are reviewed, with a focus on their application to MAX phases. The article concludes with a discussion on the potential of MAX phases in various technological applications and the importance of future research in this field.This article provides a critical review of the M$_{n+1}$AX$_n$ phases, known as MAX phases, from a materials science perspective. MAX phases are a class of hexagonal-structured ternary carbides and nitrides (X) formed by a transition metal (M) and an A-group element. The most well-known MAX phases include Ti$_2$AlC, Ti$_3$SiC$_2$, and Ti$_4$AlN$_3$. These phases exhibit unique combinations of chemical, physical, electrical, and mechanical properties, making them potentially useful in various applications such as high-temperature structural materials, protective coatings, sensors, and low-friction surfaces. 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 by epitaxial growth. However, the surface-initiated decomposition of MAX phases into MX compounds at temperatures of 1000–1100 °C is lower than the decomposition temperatures typically reported for bulk materials. Recent advancements in low-temperature synthesis, such as the deposition of V$_2$GeC and Cr$_2$AlC at ~450 °C, have also been discussed. The article reviews the progress in first-principles calculations for predicting hypothetical MAX phases and their properties, leading to the discovery of new MAX phases like Ti$_4$SiC$_3$, Ta$_4$AlC$_3$, and Ti$_3$SnC$_2$. Future research directions include charting unknown regions in phase diagrams, understanding the effects of anisotropy, impurities, and vacancies on electrical properties, and exploring unexplored properties such as superconductivity, magnetism, and optics. The article also covers the fundamentals of MAX phases, including their crystal structure, phase diagrams, and terminology. It discusses polymorphism, intergrown structures, vacancies, and solid solutions in MAX phases. The thin-film processing methods, such as physical vapor deposition (PVD), chemical vapor deposition (CVD), and solid-state reaction synthesis, are reviewed, with a focus on their application to MAX phases. The article concludes with a discussion on the potential of MAX phases in various technological applications and the importance of future research in this field.