Summary: This paper is a survey about some recent results obtained by the authors on the so-called principal curvature theorem and its applications to the study of the curvature of complete oriented hypersurfaces in Euclidean space with constant mean curvature and, more generally, with constant higher order mean curvature. The original results can be found in the following joint papers [the first and the second author, J. Math. Anal. Appl. 363, No. 2, 579–587 (2010; Zbl 1182.53052); Geom. Dedicata 156, 31–47 (2012; Zbl 1232.53046); the first and the third author, Geom. Dedicata 182, 117–131 (2016; Zbl 1343.53054); Geom. Dedicata 199, 273–280 (2019; Zbl 1430.53060)].