In mathematical fluid dynamics the Boussinesq equations constitute a widely used model of motion of a viscous, incompressible buoyancy-driven fluid flow coupled with heat convection. This thesis deals with existence and uniqueness of weak solutions of the Boussinesq equations with no slip boundary condition for the velocity field uand Robin boundary condition for the temperature thetain domains satisfying a uniform Lipschitz condition. We investigate the influence of surface roughness to weak solutions of the Boussinesq equations. The main tool of our approach is the theory of Young measures. Optimizing the heat transfer is an important application of fluid mechanics. We address the problem of optimizing the surface roughness such that the heat energy transferred through the boundary becomes maximal/minimal. Furthermore, we deal with existence and uniqueness of strong solutions of the Boussinesq equations with Dirichlet boundary conditions for u, theta in arbitrary domains Omega subseteq R3.These results will be used to formulate regularity criteria for weak solutions of the Boussinesq equations.