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The purpose of this thesis is to report a theoretical investigation on the current-current correlation and noise in the tmnsient quantum transport regime. In particular, we calculate current correlations when the bias voltage of a LDL quantum device is suddenly turned off. Namely, we consider the situation that when time t < 0 the device is in a steady-state under bias Vb, when t > 0 the bias is turned off to zero. Under such a bias, the transport current l goes from a finite steady-state value 10 at t < 0 to zero at large times. When electronic structure of the leads as well as well as the device scattering region are to be taken into account, it is a difficult problem to calculate the time dependent current-current correlation. However, for the sharp step-down bias shape, we discover that the time-dependent problem can be solved exactly for non-interacting systems.