We present a comprehensive review of implementation and application of Laplace deep-leve1 transient spectroscopy (LDLTS). The various approaches that have been used previously for high-resolution DLTS are outlined and a detailed description is given of the preferred LDLTS method using Tikhonov regularization. The fundamental limitations are considered in relation to signal-to-noise ratios associated with the measurement and compared with what can be achieved in practice. The experimental requirements are discussed and state of the art performance quantified. The review then considers what has been achieved in terms of measurement and understanding of deep states in semiconductors through the use of LDLTS. Examples are given of the characterization of deep levels with very similar energies and emission rates and the extent to which LDLTS can be used to separate their properties. Within this context the factors causing inhomogeneous broadening of the carrier emission rate are considered. The higher resolution achievable with LDLTS enables the technique to be used in conjunction with uniaxial stress to lift the orientational degeneracy of deep states and so reveal the symmetry and in some cases the structural identification of defects. These issues are discussed at length and a range of defect states are considered as examples of what can be achieved in terms of the study of stress alignment and splitting. Finally the application of LDLTS to alloy systems is considered and ways shown in which the local environment of defects can be quantified.