We present a detailed study of the possibility of having a horizontal gauge symmetry, in addition to SU(3)c × SU(2)L × U(1). Grand unification is not used as a constraint. We concentrate on the horizontal group SU(3)H. There are two main results. (i) The horizontal symmetry might break on two mass scales, one heavy and the other of order mz or a few mz. The KL-KS mass difference and other rare kaon and muon transitions are protected by a residual symmetry, but other flavor changing processes not involving the lightest fermion of each charge can be orders of magnitude larger, e.g. b →sℓ+ℓ−, τ → μℓ+ℓ−,e+e− →bs¯, τμ,tc¯(while b→dℓ+ℓ−, τ →eℓ+ℓ− and some others are forbidden). Some sectors should have relatively large flavor changing neutral currents. (ii) There is not enough new physics in the approach to explain fermion masses, but the structure of the model suggests that the lighter masses and the KM quark mixing angles might be generated radiatively. Unfortunately, careful analysis shows that the same symmetry that protects the KL-KS mass difference requires that the Cabibbo angle be identically zero to all orders. This suggests that it may be extremely difficult to make a realistic fundamental or effective SU(3) horizontal symmetry. While the model fails to generate the CKM angles, it is consistent with all low energy data and suggests a number of interesting reactions that might occur.