The Local Discontinuous Galerkin (LDG) method that is presented here provides a unified framework and an elegant approach for solving various kinds of heat conduction problems like thermal contact resistance and sharp/high gradient problems without much modifications to the basic formulation. In this paper, we describe the LDG formulation for parabolic heat conduction problems. The advantages of LDG method over the Continuous Galerkin (CG) finite element method are shown using two classes of problems - problems involving sharp/high gradients, and imperfect contact between surfaces. So far, interface/gap elements have been primarily used to model the imperfect contact between two surfaces to solve thermal contact resistance problems. The LDG method eliminates the use of interface/gap elements and provides a high degree of accuracy. It is further shown in the problems involving sharp/high gradient, that the LDG method is less expensive (requires less number of degrees of freedom) as compared to the CG method to capture the peak value of the gradient. Several illustrative 1-D/2-D applications highlight the effectiveness of the present LDG formulation.