The steady, laminar, incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are investigated. The governing Navier-Stokes and thermal equations of the flow are reduced to a nonlinear system of ordinary differential equations. The equations are solved analytically using the homotopy analysis method (HAM). Convergence of the HAM solutions is discussed in detail. These solutions are then compared with recently obtained numerical and perturbative solutions. Plots of the velocity and temperature profiles are provided for various values of the relevant parameters.