We present the results of an all-sky, deep optical survey for faint Local Group dwarf galaxies. Candidate objects were selected from the second Palomar Observatory Sky Survey and ESO/Science Research Council survey plates, and follow-up observations were performed to determine whether they were indeed overlooked members of the Local Group. Only two galaxies (Antlia and Cetus) were discovered this way out of 206 candidates. Based on internal and external comparisons, we estimate that our visual survey is more than 77% complete for objects larger than 1' in size and with a surface brightness greater than an extremely faint limit over the 72% of the sky not obstructed by the Milky Way. Our limit of sensitivity cannot be calculated exactly, but it is certainly fainter than 25 mag arcsec^-2 in R, probably 25.5 and possibly approaching 26. We conclude that there are at most one or two Local Group dwarf galaxies fitting our observational criteria still undiscovered in the clear part of the sky, and roughly a dozen hidden behind the Milky Way. Our work places the "missing satellite problem" on a firm quantitative observational basis. We present detailed data on all our candidates, including surface brightness measurements.