The structural properties of Scandium Carbide (ScC) and Niobium Carbide (NbC) were studied in five crystalline structures: rock salt (NaCl), cesium chloride (CsCl), zinc-blende (ZB), wurtzite (WZ), and nickel arsenide (NiAs). Calculations were based on the full-potential linearized plane wave method (FP-LAPW) with the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof, in the framework of density functional theory (DFT). We found that the ground states are AsNi-like and NaCl-like phases for ScC and NbC respectively, although it is possible that ScC may exist in the NaCl phase and that the NbC may exist in the AsNi one. In the cubic phases, the lattice constants and the bulk modulus are in good agreement with other research. Also, we found that ScC changes from the NaCl or NiAs to the CsCl structure at similar to 127 GPa with a volume reduction of similar to 8.5 % and similar to 7.4 % respectively; therefore, the compound can coexist in three phases: NiAs, NaCl and CsCl. A phase transformation from ZB to CsCl at similar to 18.3 GPa was found for NbC. The density of states (DOS) shows that both compounds are non-magnetic and exhibit metallic behavior.