  We prove that an $L^\infty$ potential in the Schr\"odinger equation in three
and higher dimensions can be uniquely determined from a finite number of
boundary measurements, provided it belongs to a known finite dimensional
subspace $\mathcal W$. As a corollary, we obtain a similar result for
Calder\'on's inverse conductivity problem. Lipschitz stability estimates and a
globally convergent nonlinear reconstruction algorithm for both inverse
problems are also presented. These are the first results on global uniqueness,
stability and reconstruction for nonlinear inverse boundary value problems with
finitely many measurements. We also discuss a few relevant examples of finite
dimensional subspaces $\mathcal W$, including bandlimited and piecewise
constant potentials, and explicitly compute the number of required measurements
as a function of $\dim \mathcal W$.
