  We describe the implementation of weak quantum measurements in
superconducting qubits, focusing specifically on transmon type devices in the
circuit quantum electrodynamics architecture. To access this regime, the
readout cavity is probed with on average a single microwave photon. Such
low-level signals are detected using near quantum-noise-limited superconducting
parametric amplifiers. Weak measurements yield partial information about the
quantum state, and correspondingly do not completely project the qubit into an
eigenstate. As such, we use the measurement record to either sequentially
reconstruct the quantum state at a given time, yielding a quantum trajectory,
or to close a direct quantum feedback loop, stabilizing Rabi oscillations
indefinitely.
