This talk will address some recent mathematical advances on two problems which occur in parameter estimation in PDEs from observed data. The first one concerns the identifiability of the scalar non-constant diffusion coefficient $a$ in a second order elliptic PDE from the observation of the full solution $u$ for a given right hand side $f$. Our main results show that for strictly positive right hand side, and $a$ belonging in certain smoothness classes, identification is possible with Hölder dependence of $a$ on $u$, and that Lipschitz dependence does not generally hold. The second problem concerns the recovery of the full solution $u$ from a finite number of linear measurements representing the observed data. Motivated by reduced modeling, the $a$-priori additional information about $u$ is in the form of how well it can be approximated by a certain known subspace of given dimension (reduced bases, POD). Algorithms that yield near optimal recovery bounds are proposed. These results were obtained in collaborations with Peter Binev, Andrea Bonito, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova, Gerrit Welper and Przemyslaw Wojtaszczyk.