We report the synthesis of CdSe/CdS semiconductor core/shell nanocrystals with very thick (5 nm) CdS shells. As in the case of core CdSe nanocrystals, we show that a thick-shell CdSe/CdS core/shell structure can be synthesized in either a pure wurtzite (W) or a zinc-blende (ZB) crystal structure. While the growth of thick-shell wurtzite CdSe/CdS is quite straightforward, we observe that the growth of a CdS shell on zinc-blende CdSe cores is more difficult and leads to wurtzite/zinc-blende polytypism when primary amines are present during the shell formation. Using absorption spectra analysis to differentiate zinc blende from wurtzite CdSe, we show that primary amines can induce a nearly complete structural transformation of CdSe ZB cores into W cores. This better understanding of the CdSe ligand-dependent crystal structural evolution during shell growth is further used to grow large (10 nm)-diameter perfect zinc-blende CdSe core crystals emitting above 700 nm, and perfect ZB thick-shell CdSe/CdS nanocrystals. We observed that all thick-shell CdSe/CdS QDs have extremely reduced blinking events compared to thin-shell QDs, without any significant influence of crystalline structure and polytypism.