In this work, we present approximate and exact solutions for the temperature profile and Nusselt number under fully developed laminar flow of a power law fluid inside pipes and between parallel plates. Constant wall temperature and negligible axial heat conduction are considered, for both the cases with and without viscous dissipation. For completeness, the corresponding solutions for the related problem of constant heat flux at the wall are also presented. In the absence of viscous dissipation, the solutions obtained are semi-analytic, since they rely upon an iterative procedure. As a benchmark result, to allow comparison with the results obtained with the semi-analytical expressions, we also present highly accurate numerical solutions for the Nusselt number, Nu, based on numerical integration of the energy equation. Also based on these numerical results, simplified correlations for Nu are proposed, valid for a wide range of the power law index.