Density conditions including necessary ones and sufficient ones for irregular wavelet systems to be frames are studied in this paper. We give a definition of Beurling density for the case of wavelet frames and show that for irregular wavelet systems to be frames, the time-scale parameters must be relatively uniformly discrete. We prove that for a nice wavelet function, every relatively uniformly discrete time-scale sequence with sufficiently high density will generate a frame. Explicit frame bounds are given. We also study the stability of wavelet frames and show that every wavelet frame with arbitrary time-scale parameters is stable provided the wavelet function is nice enough. Explicit stability bounds are given. Numerical examples show that our results are sharper than some known ones.