The CVBEM for elasticity problems in its basic formulations (direct, indirect and displacement discontinuity (DD) ones) is presented. It is based on the use of complex fundamental solutions. The complex boundary integral equations (CBIEs) arising from the basic CVBEM formulations are considered. It was shown that the developed theory includes the main CBIEs obtained by using the different approaches. Besides, new real and complex integral equations were obtained. The revealed links between real variables BEM (RVBEM) and CVBEM serve to mutual enrichment of these two approaches.