A data strobe signal generator includes a control unit, a pulse delay unit, a clock generator, and a data strobe output unit. The control unit generates a CAS latency signal and a preamble signal. The pulse delay unit delays a pulse signal for predetermined time and outputs a delayed pulse signal. The clock generator outputs a control clock signal. The data strobe output unit outputs a data strobe signal. The data strobe signal generator and the semiconductor memory device having the same generate a data strobe signal based on an adjustable preamble value, thereby ensuring the stabilized data output operation of a high-speed memory device.