Time-domain state-space inversion methods for non-dispersive layered media have shown that acoustic impedance as a function of traveltime can be determined from a normally incident, plane, pressure-wave source. The state-space method is used here to determine the density and acoustic velocity separately as functions of depth by simultaneously inverting surface data measured at two precritical angles of incidence. The acoustic state-space method is also applied to isotropic elastic media with normally incident P-waves and obliquely incident SH-wave sources, whereby the density, shear velocity, and compressional velocity are determined as functions of depth. The same parameters are recovered by modifying Shiva and Mendel's (1983) method for a single, obliquely incident P-wave source to accommodate a single SV-wave source, two P-wave sources, or data obtained from a previous (scalar) SH-wave inversion. Using data from multiple experiments eliminates nonuniqueness in the inversion procedure.We have extended the state-space approach to generally anisotropic media to solve for the layer thicknesses, densities, elastic stiffness parameters, and symmetry axis rotation angles in each layer. Three-component recording is required to identify the wave types at each interface and to downward continue the wave fields properly at depth. For transverse isotropy and all higher anisotropies, an n-parameter search is required at each interface, where 1 < or = n < or = 25. This adds an inherent nonuniqueness to the inversion. However, for transverse isotropy, nonuniqueness does not appear to be a serious problem since the layer parameters are determined by a single-parameter search of a monotonic function at each interface. It appears that slant stacking of multicomponent surface seismograms is applicable to generally anisotropic media and that methods that deal with noise and band limitations in acoustic media can be useful to anisotropic inversion as well.