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This monograph presents new spherical mean-value relations for classical boundary-value problems of mathematical physics. The derived spherical mean value relations provide equivalent integral formulations of original boundary-value problems. Direct and converse mean-value theorems are proved for scalar elliptic equations (the Laplace, Helmholtz and diffusion equations), parabolic equations, high-order elliptic equations (biharmonic and metaharmonic equations), and systems of elliptic equations (the Lame equation, systems of diffusion and elasticity equations). In addition, applications to the random-walk-on-spheres method are given. The book should be of interest to postgraduate researchers in the field of partial differential and integral equations, numerical mathematics and applied probability, as well as mathematical physics.