The more than 60 ternary carbides and nitrides, with the general formula Mn+1AXnwhere n = 1, 2, or 3; M is an early transition metal; A is an A-group element (a subset of groups 1316); and X is C and/or Nrepresent a new class of layered solids, where Mn+1Xnlayers are interleaved with pure A-group element layers. The growing interest in the Mn+1AXnphases lies in their unusual, and sometimes unique, set of properties that can be traced back to their layered nature and the fact that basal dislocations multiply and are mobile at room temperature. Because of their chemical and structural similarities, the MAX phases and their corresponding MX phases share many physical and chemical properties. In this paper we review our current understanding of the elastic and mechanical properties of bulk MAX phases where they differ significantly from their MX counterparts. Elastically the MAX phases are in general quite stiff and elastically isotropic. The MAX phases are relatively soft (28 GPa), are most readily machinable, and are damage tolerant. Some of them are also lightweight and resistant to thermal shock, oxidation, fatigue, and creep. In addition, they behave as nonlinear elastic solids, dissipating 25% of the mechanical energy during compressive cycling loading of up to 1 GPa at room temperature. At higher temperatures, they undergo a brittle-to-plastic transition, and their mechanical behavior is a strong function of deformation rate.