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The continuous and discrete PT-symmetric nonlocal nonlinear Schrödinger (NNLS) models are both found to be integrable in the sense of admitting the Lax pair and an infinite number of conservation laws. By using the Darbouxtransformation and starting from a plane-wave seed, we construct a chain of nonsingular exponential and rational soliton solutions of the defocusing NNLS model. Via the asymptotic analysis, we show that the N-th iterated exponential solutions in general display a variety of elastic interactions among2N solitons, and each interacting soliton could be either the dark or anti-dark type; and that the asymptotic solitons of higher-order (N>=2) rational solu-tions have the non-constant propagation velocities. With numerical simula-tions, we examine the stability of the exponential and rational soliton solutions when the initial values have a small shift from the center of the PT symmetry. Also, we find that the discrete NNLS model of the defocusing type admits the nonsingular rational soliton solutions like the continuous case. In addition,we obtain the nonsingular bright- and dark-soliton solutions of the focusing NNLS model and the related parametric conditions.