Given a field F of arbitrary characteristic and an algebraic torus T/F we calculate degree 2 and 3 cohomological invariants of T with values in Q/Z(1) and Q_p/Z_p(2) respectively, the latter for p not equal to 2, char(F) and generalize the former to other algebraic groups. Moreover, we obtain descriptions of the corresponding unramified cohomology groups, and in particular of H³_nr(F(T), μ_n^{\otimes 2}) for n prime to 2 and char(F). In the process, we construct a useful short exact sequence for cohomological invariants and make connections with recent results on Chow groups of codimension 2.