In order to shed light on the unexplored properties of the ternary nitride CaTiN, we report for the first time the results of an ab initio study of its structural, electronic, elastic, chemical bonding and thermodynamic properties. Calculated equilibrium structural parameters are in excellent concordance with available experimental data. Electronic properties were explored through the calculation of the energy band dispersions and density of states. It is found that CaTiN has an indirect band gap (Z-Γ) of 1.625 (1.701) eV using LDA (GGA). Nature of the chemical bonding was studied via Mulliken population analysis and charge density distribution map. It is found that the Ca-N bond is dominantly ionic, whereas the Ti-N one is dominantly covalent. Elastic properties of both single-crystal and polycrystalline phases of the title compound were explored in details using the stain-stress approach. Analysis of the calculated elastic moduli reveals that the title compound is mechanically stable, ductile and elastically anisotropic. Temperature and pressure dependencies of the unit-cell volume, bulk modulus, heat capacities, volume thermal expansion coefficient, Grüneisen parameter and Debye temperature were investigated based on the quasiharmonic Debye model.