In an inverse boundary value problem one is interesting in determining the internal properties of a medium by making measurements on the boundary of the medium. In mathematical terms, one wishes to recover the coefficients of a partial differential equation inside the medium from the knowledge of the Cauchy data of solutions on the boundary. These problems have numerous applications, ranging from medical imaging to exploration geophysics. We shall discuss some recent progress in the analysis of inverse boundary problems, starting with the celebrated Calderon problem, and point out how the methods of microlocal and harmonic analysis can be brought to bear on these problems.  In particular, inverse problems with rough coefficients and with measurements performed only on a portion of the boundary will be addressed.