In this paper, a study is made of the unsteady laminar boundary-layer equations for free convection on vertical plates and cylinders to establish necessary and sufficient conditions under which similarity solutions are possible. On the basis of these conditions, all possible cases are derived, including those for unsteady conditions. And it is found that the available similarity solutions in the literature have essentially covered all steady cases, except that the present analysis shows that somewhat more general conditions may also be taken in account.