We study the relaxation of the exciton spin (longitudinal spin-relaxation time T 1 ) in single asymmetrical quantum dots (QD) at low temperatures. The main relaxation mechanism is due to the exciton spin-acoustic phonon coupling via the strain-dependent exchange interaction; piezoelectric coupling is less efficient. For zero magnetic field, relaxation within the radiative exciton-doublet for typical QDs is very slow compared to the exciton lifetime. Relaxation to nonradiative states becomes important for QDs with large singlet-triplet splitting of a few meVs. The calculated relaxation times strongly decrease in high magnetic fields.1 Introduction Most concepts for quantum information processing involving quantum dots (QDs) are based on optical generation, manipulation and read-out of spins. One has therefore to investigate the dynamics of the excitonic spin. Studies of the longitudinal spin-relaxation time T 1 are rare since they require strict resonant excitation conditions and/or high magnetic fields. Recent resonant excitation experiments on InAs QDs indicate no exciton-spin relaxation at low temperatures and magnetic fields up to 8 T [1]. Similar results are reported for CdSe QDs with large lateral dimensions (comparable to twice the exciton Bohr radius of $5 nm in bulk CdSe crystals) [2,3]. However, longitudinal spin-relaxation times comparable to the exciton lifetime were deduced from high magnetic field experiments for much smaller CdSe QDs [4].In a previous communication [5], we studied the exciton-spin relaxation in QDs within the radiative doublet of the ground (heavy-hole) exciton state at low temperatures. In this paper we carry on these studies and investigate additionally the spin relaxation to near by nonradiative states. We discuss explicitly quantum dots based on (In, Ga)As and (Cd, Zn)Se semiconductor compounds. Besides, we consider the contribution of piezoelectric coupling to the excitonic relaxation processes which is considered as an important mechanism for electron spin relaxation in QDs [6].