A systematic investigation on the structural, elastic and anisotropic properties of B19' martensite CuZr is conducted using first principles calculations based on density-functional theory. Through geometry optimization, we observed an obvious structure change featuring a variation of the monoclinic angle beta which shifts B19' CuZr to a B33 type. In addition, the independent elastic constants of CuZr are determined by the stress-strain method, further enabling predictions of their Cauchy pressure, mechanical stability, anisotropic factor and Debye temperatures. Their anisotropic characters of linear compressibility, Young's modulus and shear modulus are analyzed in details by constructing three-dimensional models. (C) 2017 Elsevier B.V. All rights reserved.