The present investigation focuses on the nature and mechanism of superconductivity in the cerium-containing ternary borides and silicides: CeRu(,3)B(,2), CeOs(,3)B(,2), and CeRu(,3)Si(,2). These are the only ternary cerium compounds where "conventional" (as opposed to heavy fermion) superconductivity has been observed to date;The closely related crystal structures (all related to the CeCo(,3)B(,2)-type, P6/mmm) enabled the study of the effect on superconductivity of transforming continuously from one system to the other by going across the pseudoternary series Ce (Ru(,1-x)Os(,x))(,3)B(,2) and CeRu(,3)- (B(,x)Si(,1-x))(,2). These pseudoternary systems were subjected to high external pressures using a standard hydrostatic pressure clamp for lower pressures (up to 20 kbar) and a specially designed diamond anvil cell for higher pressures (up to 66 kbar). Trends are presented for the variation of the superconducting transition temperatures with changes in external pressure;Results are also included from measurements of static magnetic susceptibility and electrical resistivity of all of the materials in the;series Ce(Ru(,1-x)Os(,x))(,3)B(,2) and CeRu(,3)(B(,x)Si(,1-x))(,2). These provide some insight into the nature of the cerium valence in these compounds; *DOE Report IS-T-1216. This work was performed under contract No. W-7405-Eng-82 with the U.S. Department of Energy.