Keywords: Inverse Problems for partial differential equations, Probe Method, Enclosure Method, Dirichlet-to-Neumann map, Analytical method, nondestructive testing, discontinuity, cavity, crack, inclusion, electrical impedance tomography, Laplace equation, inverse acoustic obstacle scattering, inverse initial boundary value problem for the heat equation, complex geometrical optics solution, inverse source problem, fundamental solution, Mittag-Leffler function, Carleman function, Runge approximation theorem, unique continuation theorem, singularity of solutions, Helmholtz equation