This study deals with the flow and heat transfer of 2-D Maxwell fluid due to linear stretching of a curved surface. The heat transfer happens when temperature is specified on the boundary as prescribed surface temperature, PST, and prescribed surface heat flux, PSHF. The radiation and dissipation effects are incorporated in the energy equation. The modeling of the curved structure for a linear stretching velocity of Maxwell fluid is developed. The fluid is electrically conducted in the presence of varying applied magnetic field. The field quantities which describe the velocity, pressure, and temperature are calculated numerically. The effects of the radius of curvature, magnetic field, and non-Newtonian parameters are investigated and explained physically. This analysis adds to the mathematical understanding of boundary-driven flows in non-Newtonian fluids and enlarges the scope of its applications in the polymer industry for curved structures.