We present the results of high level ab initio calculations for the structure, harmonic and anharmonic spectroscopic constants, and ring puckering barrier of cyclobutane (C4H8) in an effort to establish the minimum theoretical requirements needed for their accurate description. We have found that accurate estimates for the barrier between the minimum (D-2d) and transition state (D-4h) configurations require both higher levels of electron correlation [MP4, CCSD(T)] and orbital basis sets of quadruple-zeta quality or larger. By performing CCSD( T) calculations with basis sets as large as cc-pV5Z, we were able to obtain, for the first time, a value for the puckering barrier that lies within 10 cm(-1) ( or 2%) from experiment, whereas the best previously calculated values were in errors exceeding 40% of experiment. Our best estimate of 498 cm-1 for the puckering barrier is within 10 cm-1 of the experimental value proposed originally, but it lies similar to 50 cm-1 higher than the revisited value, which was obtained more recently using different assumptions regarding the coupling between the various modes. It is therefore suggested that revisiting the analysis of the experimental data might be warranted. Our best computed values (at the CCSD(T)/aug-cc-pVTZ level of theory) for the equilibrium structural parameters of C4H8 are r(C- C) = 1.554 angstrom, r(C- H-alpha) = 1.093 angstrom, r(C-H-beta) = 1.091 angstrom, phi(C- C- C) = 88.1 degrees, alpha(H-alpha-C-H-beta) = 109.15 degrees, and theta = 29.68 degrees for the puckering angle. We have found that the puckering angle theta is more sensitive to the level of electron correlation than to the size of the basis set for a given method. We furthermore present anharmonic calculations that are based on a second-order perturbative evaluation of rovibrational parameters and their effects on the vibrational spectra and average structure. We have found that the anharmonic calculations predict the experimentally measured fundamental band origins within 1% (<= 30 cm(-1)) for most vibrations. The results of the current study can serve as a guide for future calculations on the substituted four-member ring hydrocarbon compounds. To this end we present a method for estimating the puckering barrier height at higher levels of electron correlation [MP4, CCSD( T)] from the MP2 results that can be used in chemically similar compounds.