Abstract. Data from the K − absorption reaction on 6,7 Li, 9 Be, 13 C and 16 O have recently been collected by FINUDA at the DAΦNE φ-factory (Laboratori Nazionali di Frascati-INFN), following an earlier lower statitics run on 12 C and some other targets. FINUDA is a high acceptance magnetic spectrometer which performed a wide range of studies by detecting the charged particles and neutrons exiting the targets after the absorption event. In this paper it is discussed about the study of the A(K − , Λp) reaction in the context of the search for deeply bound K-nuclear states. The observation of a bump in the Λp invariant mass distribution is discussed in terms of a possible signature of a deeply bound K − pp kaonic cluster as well as of more conventional physics. An overview of the experimental situation in this field will be given.