This article is a critical review of the M n+1 AX n phases ("MAX phases", where n =1, 2, or 3) from a materials science perspective. MAX phases are a class of hexagonal-structure ternary carbides and nitrides ("X") of a transition metal ("M") and an A-group element. The most well known are Ti 2 AlC, Ti 3 SiC 2 , and Ti 4 AlN 3 . There are ~60 MAX phases with at least 9 discovered in the last five years alone. What makes the MAX phases fascinating and potentially useful is their remarkable combination of chemical, physical, electrical, and mechanical properties, which in many ways combine the characteristics of metals and ceramics. For example, MAX phases are typically resistant to oxidation and corrosion, elastically stiff, but at the same time they exhibit high thermal and electrical conductivities and are machinable. These properties stem from an inherently nanolaminated crystal structure, with M n+1 X n slabs intercalated with pure A-element layers. The research on MAX phases has been accelerated by the introduction of thin-film processing methods.Magnetron sputtering and arc deposition have been employed to synthesize single-crystal material by epitaxial growth, which enables studies of fundamental materials properties. However, the surface-initiated decomposition of M n+1 AX n thin films into MX compounds at temperatures of 1000-1100 °C is much lower than the decomposition temperatures typically reported for the corresponding bulk material. We also review the prospects for low-temperature synthesis, which is essential for deposition of MAX phases onto technologically important substrates. While deposition of MAX phases from the archetypical Ti-Si-C and Ti-Al-N systems typically require synthesis temperatures of ~800 °C, recent results have demonstrated that V 2 GeC and Cr 2 AlC can be deposited at ~450 °C. Also, thermal spray of Ti 2 AlC powder has been used to produce thick coatings. We further treat progress in the use of first-principles calculations for predicting hypothetical MAX phases and their properties. Together with advances in processing and materials 3 analysis, this progress has led to recent discoveries of numerous new MAX phases such as Ti 4 SiC 3 , Ta 4 AlC 3 , and Ti 3 SnC 2 . Finally, important future research directions are discussed. These include charting the unknown regions in phase diagrams to discover new equilibrium and metastable phases, as well as research challenges in understanding their physical properties, such as the effects of anisotropy, impurities, and vacancies on the electrical properties, and unexplored properties such as superconductivity, magnetism, and optics. 4