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The purpose of this survey is to present some approximation and shape preserving properties of the so-called nonlinear (more exactly sublinear) and positive, max-product operators, constructed by starting from any discrete linear approximation operators, obtained in a series of recent papers jointly written with B. Bede and L. Coroianu. We will present the main results for the max-product operators of: Bernsteintype, Favard-Szász-Mirakjan-type, truncated Favard-Szász-Mirakjantype, Baskakov-type, truncated Baskakov-type, Meyer-König and Zellertype, Bleimann-Butzer-Hahn-type, Hermite-Fejér interpolation-type on Chebyshev nodes of first kind, Lagrange interpolation-type on Chebyshev knots of second kind, Lagrange interpolation-type on arbitrary knots, generalized sampling-type, sampling sinc-type, Cardaliaguet-Euvrard neural network-type.

Mathematics Subject Classification (2010): 41A30, 41A25, 41A29, 41A20,41A35, 41A05, 94A20, 94A12, 92B20.

Keywords: Degree of approximation, shape preserving properties, nonlinear max-product operators of: Berstein-type, Hermite-Fejér and Lagrange interpolation-type (on Chebyshev, Jacobi and equidistant nodes), Whittaker (sinc)-type, sampling-type, neural network Cardaliaguet-Euvrard-type.