This book is intended to give an updated overview on the state-of-the art of the theoretical and experimental efforts aimed to detect the elusive Lense-Thirring effect in the gravitational field of the Earth. The reader, after a robust introduction to the historical (Chapter 2) and theoretical (Chapters 3-5) aspects of the subject, will get acquainted with the subtleties required to design suitable observables which are able to sufficiently enhance the signal-to-noise ratio. Moreover, he/she should be able to follow autonomously the exciting developments which, hopefully, will take place in the near future if and when reliable few percent tests of this prediction of general relativity should become available. In an Earth-space based experiment with artificial satellites a good compromise between the need of reducing the impact of the systematic errors of gravitational origin and of non-gravitational origin must be obtained; this is not an easy task because such requirements are often in conflict one with each other. Consequently, a great attention is paid to elucidate many classical perturbing effects which, if not carefully modelled and accounted for in the data analysis, may alias the recovery of the gravitomagnetic signature. Indeed, we are dealing with a fundamental test of general relativity which must be honest, robust and based on solid error analysis. A critical and detailed discussion of the latest test with the LAGEOS satellites is included. The book will also be useful for better understanding the interplay among various geodetic, geophysical, general relativistic, astronomical and matter-wave interferometric effects which occurs in the weak-field and slow-motion approximation and which will become increasingly important in the near future thanks to the improvements in the accuracy of the orbital reconstruction process.
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