We define for $mathbb{R}^kappa$-Anosov actions a notion of joint Ruelle resonance spectrum by using the techniques of anisotropic Sobolev spaces in the cohomological setting of joint Taylor spectra. We prove that these Ruelle-Taylor resonances are intrinsic and form a discrete subset of $mathbb{C}^kappa$ and that $0$ is always a leading resonance. The joint resonant states at $0$ give rise to measures of SRB type and the mixing properties of these measures are related to the existence of purely imaginary resonances. The spectral theory developed in this article applies in particular to the case of Weyl chamber flows and provides a new way to study such flows.
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