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This paper reviews our recent study on a coherent optical phonon in a hexagonal YMnO₃ thin film together with related optical studies in hexagonal <i>R</i>MnO₃ (<i>R</i> = Y, Lu, Ho) compounds. Coherent phonons have been observed in <i>R</i>MnO₃ compounds by pump-probe spectroscopy with subpicosecond laser pulses, whereas the observation of coherent optical phonons was reported only in LuMnO₃. Recently, we succeeded in the observation of the coherent optical phonon in a YMnO₃ thin film. The generation process of the coherent optical phonon is assigned to a displacive mechanism, which is identical to that in LuMnO₃. The coherent optical phonon is observed in the temperature range from 10 K to room temperature, while the oscillation intensity strongly decreases as the temperature increases to the N&#233;el temperature of ~70 K from a lower temperature range. It is interesting that the temperature dependence is largely different from that in LuMnO₃. We describe that the result can be qualitatively explained by the property of an isostructural transition around the N&#233;el temperature in <i>R</i>MnO₃ compounds. In addition, we briefly discuss ultrafast incoherent responses of excited electronic states from the viewpoint of the excitation photon energy of laser pulses.