  In this review, we discuss recent experiments that investigate how the
quantum sate of a superconducting qubit evolves during measurement. We provide
a pedagogical overview of the measurement process, when the qubit is
dispersively coupled to a microwave frequency cavity, and the qubit state is
encoded in the phase of a microwave tone that probes the cavity. A continuous
measurement record is used to reconstruct the individual quantum trajectories
of the qubit state, and quantum state tomography is performed to verify that
the state has been tracked accurately. Furthermore, we discuss ensembles of
trajectories, time-symmetric evolution, two-qubit trajectories, and potential
applications in measurement-based quantum error correction.
