Polynomials and splines are frequently used as approximating functions in curve fitting. This paper discusses curve fitting with the more general piecewise polynomials employed as the approximating functions. Various representations of piecewise polynomials are discussed and one particular form is selected as providing a basis for a well-conditioned formulation of the least-squares curve-fitting problem. Continuous and discontinuous approximations with fixed and free knots are considered from the points of view of existence, uniqueness and characterization of solutions. Two general algorithms are suggested for the continuous case with arbitrary degree and continuity with fixed knots. The imposition of end conditions is shown to be straightforward. Using the principles of dynamic programming a method is proposed that enables global solutions to be obtained economically in the case of discontinuous approximating functions with free knots.