A method is described for determining the horizontal field of velocity and velocity gradients in the Bolivian Andes in the vicinity of the Bolivian orocline, using the following data: (1) space-geodetic measurements of crustal motions; (2) the style, distribution, and rate of late Miocene-Quaternary folding and faulting; and (3) paleomagnetic measurements of rigid body rotations about a vertical axis. These data were analyzed in a network of 26 triangles which spanned the Bolivian Andes by solving the velocity gradient compatibility equations between adjacent triangles, subject to the input data constraints. This yielded 120--140 linear equations to constrain 108 unknowns, making the problem moderately overdetermined. A series of experiments were carried out with different combinations of the input data to determine the best fit, in the least squares sense, field of Quaternary horizontal velocities and velocity gradients. The azimuths of velocity vectors in the high Andes are subparallel to both the relative plate convergence vector and the symmetry axis of the Bolivian orocline, with magnitudes relative to stable South America in the range 7--17 mm yr-1, accommodating up to 30% of the relative plate convergence. However, the azimuths of velocities along the range front are nearly orthogonal to the range, with a component of distributed range-parallel dextral or sinistral shear within the Eastern Cordillera and Altiplano. The horizontal velocity field requires bending about a vertical axis of both limbs of the Bolivian orocline. The field of horizontal dilatation, assuming homogeneous deformation of the entire crustal thickness, requires vertical velocities of material points up to 2.5 mm/yr in the sub-Andean zone, decreasing to <0.1 mm/yr farther east in the Eastern Cordillera and Altiplano. The direction of maximum gradients of buoyancy stress, determined from the field of gradients of appropriate combinations of velocity gradients, using a thin sheet model of viscous flow, are broadly parallel to the direction of maximum topographic gradients in the Eastern Cordillera, providing support for the idea that the bulk behavior of the Bolivian lithosphere is fluid-like. A comparison of these gradients with buoyancy stress gradients calculated from crustal thickness contrasts, suggest that the bulk lithospheric viscosity is in the range 0.5--1.0¿1022 Pa s; gradients of velocity gradients along the eastern margin of the Bolivian Andes can be interpreted in terms of marked gradients of viscosity, with viscosity increasing toward the Brazilan shield in the cratonic core of South America. Buoyancy stresses calculated from crustal thickness contrasts, calibrated with the crustal thicknesses at the changeovers from thrust to strike-slip or normal faulting, suggest that the average lithospheric-scale deviatoric stresses are in the range 7--15 MPa. ¿ 2001 American Geophysical Union