We study temperature-aware scheduling problems under the model introduced by Chrobak et al. in [9], where unit-length jobs of given heat contributions are to be scheduled on a set of parallel identical processors. We consider three optimization criteria: makespan, maximum temperature 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 polynomial 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.