Coefficient Inverse Problems are nonlinear. The conventional method of solving them numerically is via the minimization of least squares functionals. However, these functionals are non convex. Therefore, they usually have multiple local minima and ravines. This makes all conventional methods both unreliable and unstable since any optimization algorithm stops at any point of any local minimum. The speaker with his coauthors have developed a radically new approach for a broad class of Coefficient Inverse Problems. They construct a Tikhonov-like functional which is globally strictly convex. The key element of this functional is the presence of the so-called Carleman Weight Function. Global convergence of the gradient projection method is proved. Numerical results, including ones with microwave experimental data, demonstrate a good performance.