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X-ray free-electron lasers, with pulse durations ranging from a few to several hundred femtoseconds, are uniquely suited for studying atomic, molecular, chemical and biological systems. Characterizing the temporal profiles of these femtosecond X-ray pulses that vary from shot to shot is not only challenging but also important for data interpretation. Here we report the time-resolved measurements of X-ray free-electron lasers by utilizing an X-band radio-frequency transverse deflector at the Linac Coherent Light Source. We demonstrate this method to be a simple, noninvasive technique with a large dynamic range for single-shot electron and X-ray temporal characterization. A resolution of less than 1 fs r.m.s. has been achieved for soft X-ray pulses. The lasing evolution along the undulator has been studied with the electron trapping being observed as the X-ray peak power approaches 100GW.