In the present work, we have investigated the structural, electronic, elastic, and thermodynamic properties of transition-metal boron-carbon compounds (TMBCs) (TM = Ti, Zr, Hf) using the first-principles calculations. The results showed that TMBCs are energetically and thermodynamically stable, and the sequence of phase stability is HfBC > ZrBC > TiBC. B-C bonds can be formed in TMBCs ceramics due to the strong hybridization between B-2p and C-2p states. The elastic anisotropies of TMBCs were illustrated by elastic anisotropy indexes, 3D surface constructions, and 2D projections, and the results indicated that the sequence of elastic anisotropy is ZrBC > TiBC > HfBC. Finally, the calculated minimum thermal conductivities, based on the Clarke's and Cahill's models, of all TMBCs are anisotropic with the sequence of ZrBC > TiBC > HfBC.