In this paper we review the present status of electron spectroscopy of Ce and Yb Kondo systems, with particular emphasis on photoemission and inverse photoemission. Our starting point is the situation in the mid-1980s, characterized by the development of a 'Kondo scenario' for the thermodynamic and spectral properties, as described in the review by Allen et al. (1986, Adv. Phys., 35, 275). That picture, based on room-temperature results and on a T=0 approach to the impurity Anderson Hamiltonian, has been generally confirmed and better defined over the past decade by a number of more elaborate experiments, and by new calculational schemes. The use of high-energy resolution and cryogenic temperatures has led to the direct observation of the characteristic low-energy excitations and has opened the way to stringent tests of the most fundamental aspect of model, namely its scaling properties. Despite the success of the Kondo approach, more work remains to be done. The improved quality of the data has revealed some quantitative discrepancies with the predictions of the model, stressing the need for more realistic theoretical schemes. Moreover, some recent experiments questioning the validity of the model have stirred a fierce controversy within the spectroscopic community. We review those conflicting results and discuss the possible reasons for the discrepancies, and how new experiments could solve the present difficulties.