In an inverse boundary problem, one seeks to determine the coefficients of a PDE inside a domain, describing internal properties, from the knowledge of boundary values of solutions of the PDE, encoding boundary measurements. Applications of such problems range from medical imaging to non-destructive testing. In this talk, starting with the fundamental Calderon inverse conductivity problem, we shall first discuss a partial data inverse boundary problem for the Magnetic Sch\"odinger operator on CTA manifolds. Next, we discuss first-order perturbations of biharmonic operators in the same geometric. Specifically, we shall present a global uniqueness result as well as a reconstruction procedure for the latter inverse boundary problem.