We study temperature-aware scheduling problems under the model introduced by Chrobak et al. in [6]. We consider a set of parallel identi- cal processors and three optimization criteria: makespan, maximum tem- perature and (weighted) average temperature. On the positive side, we present polynomial time approximation algorithms for the minimization of the makespan and the maximum temperature, as well as, optimal poly- nomial time algorithms for minimizing the average temperature and the weighted average temperature. On the negative side, we prove that there is no ( 4/3 - ϵ)-approximation algorithm for the problem of minimizing the makespan for any ϵ > 0, unless P = NP.