Abstract: Recent measurements by CREMA Collaboration in Paul Scherrer Institute (Switzerland) determined the proton radius in Lamb shift spectroscopy of muonic hydrogen with a significantly improved precision. However, they discovered that this determination differs from the well-accepted CODATA value by 5.6 standard deviation. This discovery is named the “proton radius puzzle”, and attracted interests of many physicists. Inspired by this work, the CREMA Collaboration extended their experiments in muonic hydrogen to a series of light muonic atoms/ions, including \mu^2,3H and \mu^3,4He^+. They planned to extract the radii of light nuclei (i.e., ^2,3H,^3,4He) from Lamb shift measurements in muonic atoms. Besides the spectroscopy precision, the accuracy of nuclear radii is limited by one theoretical input, i.e., nuclear polarizability. Nuclear polarizability originates from virtual excitation of the nucleus during the two-photon exchange process. This effect can make higher-order corrections to the muonic atom spectrum. Polarizability is strongly connected with the photonuclear reaction and the virtual Compton scattering. Therefore, its correction to the Lamb shift can be obtained by evaluating the sum rules of photoabsorption cross sections and forward virtual Compton amplitudes. Using ab initio methods, we calculated the nuclear polarizability effects in muonic atoms. By utilizing modern nuclear force models and the hyperspherical harmonic many-body approaches, we calculated a series of photonuclear sum rules, which are correlated with the nuclear polarizability. This theoretical work provides key input to the high-precision determination of nuclear radii in muonic atom spectroscopy.