SpecMAT aims at providing crucial experimental information to answer key questions about the structure of atomic nuclei:- What are the forces driving the shell structure in nuclei and how do they change in nuclei far from stability?- What remains of the Z = 28 and N = 50 “magic numbers” in 78Ni?- Do we understand shape coexistence in nuclei, and what are the mechanisms controlling its appearance?The position of natural and “intruder” shells will be mapped in two critical regions, the neutron-rich nuclei around Z = 28 and the neutron-deficient nuclei around Z = 82. The centroids of the shell strength are derived from the complete spectroscopy of those systems in nucleon-transfer measurements. This method will be applied for the first time in the region of neutron-deficient Pb nuclei.In SpecMAT (Spectroscopy of exotic nuclei in a Magnetic Active Target) a novel instrument will overcome the present challenges in performing such measurements with very weak beams of unstable nuclei. It combines high luminosity, high efficiency and a very large dynamic range and allows detection of both charged-particle and gamma-ray radiation. The instrument owns its remarkable performances to a number of advanced technologies concerning the use of electronics, gaseous detectors and gamma-ray detectors in a magnetic field.The SpecMAT detector will be coupled to the HIE-ISOLDE facility for the production and post-acceleration of radioactive ion beams in construction at CERN in Geneva. HIE-ISOLDE will provide world-unique beams thanks to the use of the proton injector of the CERN complex.If successful, SpecMAT at HIE-ISOLDE will produce specific results in nuclear structure which cannot be reached by other programmes elsewhere. Such results will have a significant impact on the present theories and models of the atomic nucleus.