Hybrid polymer-based solar cells (HPSCs) that use conjugate polymers as electron donor (D) and inorganic semiconductor nanocrystals as electron acceptor (A) are novel photovoltaic devices. HPSCs integrate the properties of organic polymer (flexibility, ease of film formation, high absorption coefficient) and inorganic nanostructures (high electron mobility, high electron affinity, and good stability), and have the extra advantages, such as the rich sources of synthesized nanostructures by wet chemistry, tunable and complementary properties of assembled components, solution-processibility on a large scale at low cost and light-weight, etc. Amongst various inorganic semiconductor materials, the nanostructured metal oxides are the promising electron acceptors for HPSCs, because they are environment-friendly, transparent in visible spectrum and easy to be synthesized. After a brief introduction to the current research status, working principles, device architecture, steady-state and dynamic characterizations of HPSCs, this paper mainly reviews our recent research advances in the HPSCs using ZnO and TiO2 nanostructures as main electron acceptor and transporter, with emphasis on the theoretical models for charge carrier transport dynamics, design and preparation of efficient materials and devices, and the device performance related with nanostructural characteristics. Finally, the main challenges in the development of efficient HPSCs in basic researches and practical applications are also discussed. The main conclusions from our studies are summarized as follows: (i) IMPS and IMVS are powerful dynamic photoelectrochemical methods for studying the charge transport dynamics in HPSCs, and our theoretical models enable the IMPS to serve as an effective tool for the mechanistic characterization and optimization of HPSC devices. (ii) Using a multicomponent photoactive layer with complementary properties is an effective strategy to achieve efficient HPSCs. (iii) Using the complementary property of components, enhancing the dissociation efficiency of excitons, and improving the transport properties of the acceptor channels with reduced energy loss to increase collection efficiency all are the effective measures to access a high photocurrent generation in HPSCs. (iv) The band levels of components in the photoactive layer of HPSCs are aligned into type II heterojunctions, in which the nanostructured component with the lowest conduction band edge acts as the main acceptor/transporter; the maximum open-circuit voltage (Voc) in HPSCs is determined by the energy difference between the highest occupied molecular orbital (HOMO) level of conjugated polymer and the conduction band edge of the main acceptor, but the Voc in practical devices correlates strongly with the quasi-Fermi levels of the electrons in the main acceptor and the holes in the polymer. While passivating the surface defects on the main acceptor, increasing spatial e-h separation, and enhancing the electron density in conduction band of the main acceptor will facilitate the increase in Voc. (v) There is no direct correlation among Voc, photogenerated voltage (Vph) and electron lifetime (τe), and they may change in the same or the opposite trend when the same or different factors affect them, therefore one should get insight into the intrinsic factors that influence them when discussing the changes in Voc, V_{ph} and τe that are subject to nanostructural characteristics.