In this work, we develop analytical homogenization estimates for the macroscopic response and field statistics of elasto-viscoplastic polycrystals in the transient and steady-state regimes. In Chapter 2, we use the iterated fully optimized second-order homogenization method for the steady-state response of porous polycrystals composed of large randomly distributed pores in a fine-grained polycrystalline matrix, with grains described by crystal viscoplasticity. The method is then used to estimate the effects of microstructure on the macroscopic response of sea ice. The key finding here is that the brine-air inclusions induce macroscopic compressibility, which is significantly affected by the porosity, pore geometry, and crystallographic texture of polycrystalline ice. Comparisons with experimental results demonstrate the capabilities of the model, especially in capturing the dilatational response of sea ice under combined hydrostatic and deviatoric loading. In Chapter 3, we generalize the model to account for the finite-strain response of porous polycrystals with pressurized pores and characterize the effects from microstructural evolution on the macroscopic response under general loading conditions. In particular, we investigate the effects of pore pressure, as well as morphological and crystallographic textures of the polycrystalline matrix, under plane strain and axisymmetric loading conditions. We find that the pore pressure can significantly harden the macroscopic response, especially for negative triaxialities, while having minimal effects for positive triaxialities. On the other hand, the crystallographic textures impact the responses of porous polycrystals for low and moderate positive triaxialities, as deduced by appropriate comparisons with porous untextured polycrystals. Moreover, different textures are developed for different loading conditions, which in turn induces significant sensitivity of the macroscopic response on loading conditions. However, for large positive triaxialities, the macroscopic response is largely controlled by the porosity evolution, with relatively weaker dependence on loading conditions. Finally, in Chapter 4, we obtain homogenization estimates for the transient response of polycrystals with Maxwell-type elasto-viscoplastic grains, exhibiting discrete relaxation spectra (short memory). However, their interaction leads to a continuous relaxation spectrum for the macroscopic response (long memory). To capture the elasto-viscoplastic interaction of the grains, and the associated long-memory effects, we develop differential variational estimates for elasto-viscoplastic polycrystals, by describing the transient response by a set of coupled nonlinear ordinary differential equations for the stress averages and fluctuation covariances in the grains. Comparisons with the Maxwellian approximation for the macroscopic response, obtained by separately homogenizing the elastic and viscoplastic components, serve to characterize the long-memory effects. We find that the long-memory effects increase with the anisotropy and nonlinearity of the viscoplastic response of the grains and that these effects are correlated with rapid changes in the inter- and intra-granular field fluctuations. To assess their accuracy, our estimates are compared with available full-field and experimental results from the literature, and good agreements are generally found even for polycrystals with highly anisotropic and nonlinear grains.