The talk will focus on the progressive development of mathematical models of the renal concentrating mechanism of the inner medulla of the kidney. The primary objective has been to make the model concentration profiles as close as possible to the currently available experimental concentration results, with parsimonious increments in model complexity, and minimum simultaneous changes to experimentally available parameters and input values. These models, based on carefully selected currently available data and histotopography, have led to salt and urea concentrations in the collecting duct that are significantly better than concentrations from our earlier models. The inverse problem will also be discussed. In this case, for a desired concentration profile, the input values, parameters and model complexity are estimated by an inverse problem algorithm.