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We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $\mathcal{L}$. Given a wavelet generating kernel g and a scale parameter $t$, we define the scaled wavelet operator $T_g^t = g(t\mathcal{L})$. The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on $g$, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing $\mathcal{L}$. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.