THE X-RAY SPECTRAL ANALYSIS OF COMPTON THICK (CT- I.E. HAVING INTRINSIC OBSCURATION NH>1E24 CM^-2) ACTIVE GALACTIC NUCLEI (AGN) IS KEY TO UNDERSTAND THE PHYSICS OF THE OBSCURING MATERIAL SURROUNDING ACCRETING SUPERMASSIVE BLACK HOLES. FOR THIS PURPOSE THE LAUNCH OF THE NUCLEAR SPECTROSCOPIC TELESCOPE ARRAY (HEREAFTER NUSTAR) THE FIRST TELESCOPE WITH FOCUSING OPTICS AT>10 KEV REPRESENTED A MAJOR BREAKTHROUGH IN THE DETECTION AND CHARACTERIZATION OF OBSCURED AGN PROVIDING AN IMPROVEMENT ON SENSITIVITY OF ABOUT TWO ORDERS OF MAGNITUDE WITH RESPECT TO PREVIOUS FACILITIES AT THESE ENERGIES. HOWEVER A SYSTEMATIC ANALYSIS OF THE ROLE OF NUSTAR IN THE CHARACTERIZATION OF HEAVILY OBSCURED AGN IS SO FAR ABSENT IN THE LITERATURE OR LIMITED TO SMALL SAMPLES OF OBJECTS. TO OVERCOME THIS WE PROPOSE A PROJECT TO FULLY CHARACTERIZE A SAMPLE OF 35 CANDIDATE CT-AGN DETECTED IN A VOLUME LIMITED SAMPLE OF BRIGHT AGN SELECTED USING SWIFT-BAT AND HAVING AN ARCHIVAL NUSTAR OBSERVATION. ALL THE PROPOSED SOURCES ALSO HAVE BEEN OBSERVED IN THE 0.5-10 KEV BAND THUS ALLOWING US TO PERFORM A SPECTRAL FITTING OVER THE 0.5-150 KEV ENERGY RANGE. WE ENVISION THAT THIS PROGRAM THANKS TO ITS MUCH LARGER STATISTICS WITH RESPECT TO PREVIOUS WORKS WILL ACHIEVE SEVERAL GOALS. THE MOST IMPORTANT ONES ARE THE FOLLOWING: 1) STUDY AND DETERMINE THE PROPERTIES AND GEOMETRY OF THE OBSCURING MATERIAL USING DIFFERENT SELF-CONSISTENT MODELS SPECIFICALLY DEVELOPED TO FIT THE X-RAY SPECTRA OF HEAVILY OBSCURED AGN. WE WILL USE THESE MODELS TO MEASURE THE COLUMN DENSITY OF THESE OBJECTS VALIDATING OR REJECTING THEIR CT- NATURE AND OBTAINING THE MOST ACCURATE MEASUREMENT OF THE INTRINSIC FRACTION OF CT-AGN IN THE LOCAL UNIVERSE. 2) STUDY THE GEOMETRY OF THE OBSCURING MATERIAL SURROUNDING THE ACCRETING SUPERMASSIVE BLACK HOLE MEASURING THE TORUS COVERING FACTOR. THE STUDY OF HOW COVERING FACTOR CORRELATES WITH PARAMETERS SUCH AS THE TORUS COLUMN DENSITY AND THE AGN X-RAY LUMINOSITY WILL OFFER FUNDAMENTAL INFORMATIONS ON THE SMBH ACCRETION PROCESS. 3) AFTER USING THE SPECTRAL FITTING RESULTS TO CONSTRAIN WITH UNPRECEDENTED QUALITY THE CT-AGN SELECTION FUNCTION USE IT TO DERIVE THE CT-AGN SPACE DENSITY. THIS REFINED SPACE DENSITY WILL ALLOW US TO PUT STRONGER CONSTRAINTS ON THE ROLE OF CT-AGN IN THE GENERATION OF THE CXB. PARTICULARLY WE WILL VERIFY IF THE KNOWN DISCREPANCY BETWEEN OBSERVATIONS AND MODELS IS CAUSED BY A TENDENCY TO OVERESTIMATE THE NUMBER OF CT-AGN USING BAT DATA ONLY. THIS PROJECT FULLY REALIZES THE POTENTIAL OF NUSTAR WHICH WAS LAUNCHED TO DETECTED AND STUDY NEW COMPTON-THICK AGN.