The evolution of superconductivity in ultrathin films of Sn, Pb, Ga, Al, and In has been examined as a function of thickness and temperature. The films were grown in increments by condensation from the vapor onto substrates held at temperatures below 18 K. For each metal, global superconductivity or zero electrical resistance was found when the normal-state sheet resistance RN fell below a value close to h/4e2, or 6.45 k/-, an observation uncorrelated with either structural or material parameters such as thickness or transition temperature. Prior evidence of superconductivity with nonzero resistance, local superconductivity, was found at earlier stages of film growth. All evidences of superconducting behavior were observed at temperatures close to the bulk transition temperature beginning in the range of thicknesses for which normal-state resistivities were greater than 200 -cm and were rapidly changing with thickness. This implies that the films consisted of fully superconducting grains connected by tunneling junctions. The strong disorder represented by a broad distribution of junction parameters can be renormalized into weak disorder. Thus theoretical calculations based on regular arrays of superconducting sites coupled by (Josephson) junctions appear to be relevant. The extreme thinness of the films implies very small junction capacitances leading to large quantum fluctuations of the phase differences of their superconducting order parameters. Two classes of theories explaining a nearly universal resistance threshold for superconductivity have emerged. Both classes involve the quenching of these quantum fluctuations. In the limit of very small junction capacitances the threshold occurs at resistance values near h/4e2, and is essentially independent of the capacitance and the energy gap, in good agreement with the experimental data. Not contained in any of the models is an explanation of the observed regular variation of the low-temperature resistances of the films with the normal-state sheet resistance for values just above the resistance threshold.