We calculate two-dimensional steady-state distributions of transport electric field $\mathbf{E}(x,y)$ and current density $\mathbf{J}(x,y)$ in superconductors with restricted geometries, such as films with macroscopic planar defects, faceted grain boundaries, current leads, flux transformers, and microbridges. We develop a hodograph method, which enables us to solve analytically Maxwell's equations for $\mathbf{E}(x,y)$ and $\mathbf{J}(x,y),$ taking account of the highly nonlinear $E\ensuremath{-}J$ characteristics of superconductors ${E=E}_{c}{(J/J}_{c}{)}^{n},n\ensuremath{\gg}1.$ Based on this approach, a very effective numerical method of solving the nonlinear Maxwell equations was also developed. We show that nonlinear current flows in restricted geometries exhibit orientational current-flow domains separated by domain walls of varying width, which remain different from the discontinuity lines of the Bean model, even in the critical state limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}.$ The nonlinearity of $E(J)$ gives rise to new length scales for $\mathbf{E}(x,y)$ and $\mathbf{J}(x,y)$ distributions, strong local enhancement of $\mathbf{E}(x,y)$ and long-range electric-field disturbances around planar defects on the scale ${L}_{\ensuremath{\perp}}\ensuremath{\sim}\mathrm{an}$ much greater than the defect size a. For instance, a planar defect of length $agd/n$ in a film of thickness d produces a narrow $(\ensuremath{\sim}d/\sqrt{n})$ magnetic-flux jet (domain of high electric field), which spans the entire current-carrying cross section. As a result, even small defects $(a\ensuremath{\sim}d/n),$ which occupy only a small fraction of the geometrical cross section, give rise to significant peaks of voltage and dissipation. This nonlinear current blockage by planar defects (high-angle grain boundaries, microcracks, etc.) essentially affects the global $E\ensuremath{-}J$ characteristics and critical currents in superconductors.