A review is presented of our recent work which deals with some aspects of the inclusion of connected triple excitations in single reference coupled–cluster theory. We first consider basic aspects of the theory and computational cost, noting the need for accurate and economical approximations to the complete coupled–cluster single, double and triple excitation method (CCSDT). Several such approximations are now available, including the CCSD + T(CCSD) and CCSD(T) methods. We have now efficiently implemented analytical derivative methods for these and related methods for open-shell reference determinants, including restricted open-shell Hartree–Fock for CCSD(T). Some results with large basis sets are presented for the HCO radical which further demonstrate the reliability of these methods for the spectroscopic characterization of small open-shell species. The remainder of this article deals with several applications of the complete CCSDT method, a new implementation of which we have recently completed. Like our previous work, this is for general single reference determinants, although we have now incorporated Abelian symmetry. We present a comparison with full configuration interaction data for CH3; a study of the N2 potential curve using restricted and unrestricted Hartree–Fock references; a comparison of the dipole moments and polarizabilities of BeO; and a study of the harmonic force field of 2Σ C. These studies also provide assessments of the accuracy of the approximate schemes for including T3. Finally, we present some timing data for some CCSDT calculations with out new implementation. © 1993 John Wiley & Sons, Inc.