This paper presents a class of boundary integral equation methods for the numerical solution of acoustic and electromagnetic time-domain scattering problems in the presence of unbounded penetrable interfaces in two spatial dimensions.The proposed methodology relies on convolution quadrature (CQ) schemes and the recently introduced windowed Green function (WGF) method. As in standard time-domain scattering from bounded obstacles, a CQ method of the user’s choice is used to transform the problem into a finite number of (complex) frequency-domain problems posed, in our case, on the domains containing unbounded penetrable interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF metho—which introduces errors that decrease super-algebraically fast as the window size increases.The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off-the-shelf Nyström or boundary element Helmholtz integral equation solvers capable of handling complex wavenumbers with large imaginary part. A high-order Nyström method based on Alpert’s quadraturerules is used here. A variety of CQ schemes and numerical examples, including wave propagation inopen waveguides as well as scattering from multiplelayered media, demonstrate the capabilities of the proposed approach.