calculated at a 95% confidence level. The experimental data were compared with an approximate theory using the PFA and with the exact theory based on the scattering approach. Note that exact calculation methods allowing the evaluation of the Casimir force between bodies of arbitrary shape and made of any real material have been actively investigated since 2006 [35, 36, 61–71]. However, until the present paper and the rapid communication [41] there was no detailed comparison of exact theoretical results beyond the Lifshitz theory taking into account such experimental conditions as real material properties of the test bodies, surface roughness and nonzero temperature with measurement data of any specific experiment. Using the exact theory, the computed dependences of the lateral Casimir force on the phase shift and of the maximum magnitude of the lateral force as a function of separation were found to be in agreement with the experimental data. Regarding the PFA approach, the computational results were found to be excluded by the data. This provides the quantitative confirmation for the fact that this experiment was performed outside the applicability region of the PFA and marks the beginning of investigations of diffraction effects in the phenomenon of the lateral Casimir force. As was already noted in the Introduction, the lateral Casimir force might find applica- tions in nanotechnology. Specifically, it was proposed to use this effect for the frictionless transmission of lateral motion by means of a nanoscale rack and pinion without intermesh- ing cogs [28, 29] or in a ratchet with asymmetric corrugations driven by the Casimir force [30]. The above experimental and theoretical results bring us closer to the realization of such kinds of micromachines. In future it is planned to perform similar experiments with corrugations of more complicated shape which provides an opportunity to modulate the respective lateral Casimir force. This work was supported by the NSF Grant No. PHY0653657 (measurement of the lateral Casimir force) and DOE Grant No. DE-FG02-04ER46131 (calculation of the electric and Casimir forces). V.N.M. and U.M. were also supported by the DARPA Grant N66001-09- 1-2069. G.L.K. and V.M.M. were partially supported by the DFG Grant GE 696/10-1. V.N.M. was partially supported by the Grants RNP 2.1.1/1575 and RFBR 07-01-00692- Acknowledgment 35