A summary is given of past attempts to find methods for deducing potential interactions from elastic scattering phase versus energy curves. The distorted plane wave approximation for tg η l ( k) of the first author is improved by introducing energy dependent terms in the wave function. The phase-potential expression may then be made as accurate as the phase data supplied and is applied to obtain a practical method of deducing potentials from η l ( k) curves. The latter need only be over a finite range of k, and η l ( k max) need not be small. Phase analysis of elastic scattering data provides smooth η l ( k) curves for particular l values, which may be least-square fitted by orthogonal polynomial series to give truncated polynomials, in powers of k 2, of k 2 l+1 cotg η l ( k) or tg η l ( k)/ k 2 l+1 . The scattering parameters (coefficients of k 2) in principle identify the interaction potential, although they are in general functions of the total energy interval 0- E max covered, as the corresponding infinite power series is divergent. The tg η l ( k) approximation reduces to the shape-dependent expansion, the scattering parameters being given as linear combinations of moment and weighted moment integrals over the interaction potential. We employ the experimental values of the scattering parameters in these relations. If the potential is expanded as a finite series of short range functions, such as the exponential, the series coefficients may be found by solving a corresponding number of simultaneous equations based on the above. The resulting approximate potential has a strength function (first moment integral of the potential) accurate to within several per cent. A simple iterative procedure, based on the distorted plane wave function and equations, is used to improve the potential, the error in the strength function being reduced by a factor of 2–5 per iteration in typical cases. This method may also be used to improve arbitrary approximate potentials, any accuracy being possible.