Forced convection heat transfer in fully developed laminar flows of power-law fluids in eccentric annular ducts is computationally analyzed. With an insulated outer surface, the heating or cooling on the inner surface is modeled by two fundamental boundary conditions - uniform axial heat flux (HI) and constant wall temperature (T) - commonly encountered in thermal processing applications. Numerical solutions for the velocity and temperature distributions, isothermal frictions factors, and Nusselt numbers for annular ducts of varying aspect ratios (0.2 less than or equal to r* less than or equal to 0.8) and inner core eccentricity (0 less than or equal to epsilon* less than or equal to 0.6) are presented for both shear-thinning (0.2 less than or equal to n < 1) and shear-thickening (1 < n less than or equal to 1.8) fluids. Due to the geometric asymmetry of the eccentric annular cross-section, the flow tends to stagnate in the narrow section and have higher peak velocities in the wide section. This induces greater non-uniformity in the temperature field, and degradation in the average heat transfer coefficient. The nonlinear shear behavior of the fluid further aggravates the flow and temperature maldistribution, which produces a significantly anomalous thermal-hydraulic performance.