We shall dedicate the first half of the talk to introduce classical Higgs bundles and describe the fibres of the corresponding Hitchin fibration in terms of spectral data. Then, we shall define principal Higgs bundles and look at some examples. Finally, we consider the particular case of U(p,p) and SU(p,p) Higgs bundles and study the spectral data. As an application of our results, we shall give a new geometric description of some of the connected components of the corresponding moduli space.