The linear elastic analysis of homogeneous, isotropic cracked bodies started in the 1900s. The existence of three dimensional corner point effects in the vicinity of a corner point where a crack front intersects a free surface was investigated in the late 1970s. An approximate solution by Bažant and Estenssoro explained some features of corner point effects but there were various paradoxes and inconsistencies. Results derived from finite element models showed that the analysis is incomplete. The stress field in the vicinity of a corner point appears to be the sum of two different singularities (i.e. stress intensity factors and corner point singularities). In this paper some recent results for the corner point singularities under in and out of plane loadings is reviewed and discussed.