This chapter is devoted to present some recent results in shape optimization problems involving heat-conductive flows. We focus on the optimal shape design of a two dimensional container to achieve a temperature distribution as uniform as possible. Our geometrical setting allows to prove the well-posedness to the Boussinesq system, the constraint, for each admissible domain. Moreover, the weak solution has \(H^{3/2}\) regularity on the boundary portion where (smooth) deformations are allowed. Further, we prove the existence of a solution to the shape optimization problem of interest. We finally present a directional differentiability result and deduce a first-order optimality condition.