We present new constraints on the couplings of axions and more generic axion-like particles using data from the EDELWEISS-II experiment. The EDELWEISS experiment, located at the Underground Laboratory of Modane, primarily aims at the direct detection of WIMPs using germanium bolometers. It is also sensitive to the low-energy electron recoils that would be induced by solar or dark matter axions. Using a total exposure of up to 448 kg.d, we searched for axion-induced electron recoils down to 2.5 keV within four scenarios involving different hypotheses on the origin and couplings of axions. We set a 95% CL limit on the coupling to photons $g_{Agamma}<2.13times 10^{-9}$ GeV$^{-1}$ in a mass range not fully covered by axion helioscopes. We also constrain the coupling to electrons, $g_{Ae} < 2.56times 10^{-11}$, similar to the more indirect solar neutrino bound. Finally we place a limit on $g_{Ae}times g_{AN}^{rm eff}<4.70 times 10^{-17}$, where $g_{AN}^{rm eff}$ is the effective axion-nucleon coupling for $^{57}$Fe. Combining these results we fully exclude the mass range $0.91,{rm eV}<m_A<80$ keV for DFSZ axions and $5.73,{rm eV}<m_A<40$ keV for KSVZ axions.
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