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Albert Einstein developed his theory of General Relativity in 1916 which, along with his theory of Special Relativity (1905), has superseded Isaac Newton's theory of gravity (1687). General relativity correctly explains the perihelion precession of Mercury's orbit, the deflection of light due to a large massive body, the gravitational redshift of light, the time delay of signals passing close to massive bodies, and the loss of energy by gravitational radiation from binary pulsars. Gravity Probe B is a satellite based experiment designed to detect gravitomagnetic and gravitoelectric effects predicted by General Relativity; more specifically, Gravity Probe B is looking for a frame-dragging and a geodetic effect. The former is a precession of vector quantities such as the spin vector, due to the rotation of the Earth in the fabric of space-time. The latter is also a precession of vector quantities; however, this precession arises solely due to the existence of a large mass, the Earth. Although Gravity Probe B is a new experiment with the goal of testing the predictions a relatively new theory, similar effects to these were considered by Michael Faraday in his search for a unified theory in the early 19th century. Currently, physics has two major and well tested theories of nature - General Relativity and the Standard Model of Particle Physics based on quantum field theory. Quantum field theory describes the world of atoms and subatomic world; it was developed in the early 1920s by Max Planck, Werner Heisenberg, Erwin Schrodinger, Paul Dirac, Richard Feynman and many other physicists. Unfortunately, these theories are not compatible with each other when gravity is strong and distances are small. This suggests that one, or both, need to be modified if physics hopes to develop a theory that can accurately describe the universe in all domains. Kaluza-Klein gravity extends the four-dimensional spacetime (three space and one time) of Einstein's theory to a five-dimensional spacetime (four space and one time). The induced matter variation of Kaluza- Klein gravity is of specific interest in this thesis. In the induced-matter theory, matter and energy in our four-dimensional world is induced entirely from the geometry of spacetime in the five-dimensional world. The standard model extension is a theory which allows for Lorentz invariance. In the gravitational sector of the Standard-Model Extension, the metric tensor guv that defines the gravitational field in General Relativity is supplemented by an additional tensor field suv. Fluctuations of this field can violate Lorentz symmetry the basis of Special Relativity. This thesis is primarily concerned with constraining these two modifying theories with the experimental data from Gravity Probe B. Improved limits are set on free parameters associated with the five-dimensional soliton and canonical metrics, generalizations of the static, spherically-symmetric Schwarzschild metric of standard four-dimensional General Relativity. For the Standard-Model Extension, new constraints are imposed on the constant coefficients suv associated with fluctuations of suv about their vacuum values in the low-velocity, weak-field limit. One of these, the time-time coefficient sTT, has not previously been constrained by any other experiment. The thesis concludes with an appendix describing a new set of Mathematica routines that has been written to check the flatness of solutions of Einstein's field equations in more than four dimensions, satisfying a condition of the induced-matter interpretation of Kaluza-Klein Gravity.