Condensed matter physics has witnessed great advancements in tools

developed to obtain the state of a system given its Hamiltonian. As

quantum devices are being rapidly developed, the converse task of

recovering the Hamiltonian of a many-body system from measured

observables is becoming increasingly important. In particular,such a

task is important for certifying quantum simulators and devices

containing many qubits. We show that local Hamiltonians can be

recovered from local observables alone, using computational and

measurement resources scaling polynomially with the system size. In

fact, to recover the Hamiltonian acting on each finite spatial domain,

only observables within that domain are required. The observables can

be measured in a Gibbs state as well as a single eigenstate;

furthermore, they can be measured in a state evolved by the

Hamiltonian for a long time, allowing to recover a large family of

time-dependent Hamiltonians. We generalize these results to the case

of open quantum systems, for which we provide a method to efficiently

recover the Lindbladian from local measurements on the steady state.

We derive an estimate for the statistical recovery error due to

approximation of expectation values using a finite number of samples,

which agrees well with numerical simulations.