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We present a comparative study of the electronic band structures of the compounds CuMO2 (M=B,Al,In,Ga) which belong to the family of delafossite transparent conductive oxides. The theoretical approaches we use are the standard local-density approximation (LDA) to density-functional theory, LDA+U, hybrid functionals, and perturbative GW on top of LDA or self-consistent Coulomb hole plus screened exchange calculations. The latter approach, state-of-the-art theoretical approach for quasiparticle band structures, predicts direct band gaps that are compatible with experimental optical gaps only after including the strong polaronic and excitonic effects present in these materials. For what concerns the so-called band-gap anomaly of delafossite compounds, we find that GW approaches yield the same qualitative trends with increasing anion atomic number as the LDA: accounting for the oscillator strength at the absorption edge is the key to explain the experimental trend. None of the methods that we applied beyond the simple LDA is in agreement with the small indirect gaps found by many early experiments. This supports the recent view that the absorption bands identified as a sign of the indirect experimental gaps are likely due to defect states in the gap and are not a property of the pristine material.