A causality preserving interpretation of the electromagnetic (EM) leaky-wave (LW) propagation in space and time is proposed for the first time. The Cagniard-deHoop (CdH) joint transform technique is applied for elucidating the relation between time-domain (TD) head waves (HWs), body waves (BWs), Cherenkov wave effects, and LWs. It is conjectured that the LW phenomenon in the TD is associated with a local maximum in the observed signal that occurs between the arrivals of the HW and BW constituents. A quantitative analysis that enables the space-time localization of the LW effect is performed theoretically and, then, illustrated via representative examples including the pulsed EM radiation from both a line source above a dielectric half-space, and narrow-slot antennas.