In this work, we demonstrate both theoretically and experimentally that non-periodic metallic gratings can become transparent for broadband terahertz waves. It is shown that broadband high transmission appears in aperiodic metallic gratings (including quasi-periodic and disordered ones), which originates from the non-resonant excitations in the grating system. Quasi-periodic and disordered metallic gratings effectively weaken and even eliminate Wood's anomalies, which are the diffraction-related characters of periodic gratings.Consequently, both the transparence bandwidth and transmission efficiency are significantly increased due to the structural aperiodicity. And an optimal condition is also achieved for broadband high transparency in aperiodic metallic gratings. Experimental measurements at terahertz regime reasonably agree with both analytical analysis and numerical simulations.Furthermore, we show that for a specific light source, for example, a line source, a corresponding non-periodic transparent grating can be also designed. We expect that our findings can be applied for transparent conducting panels, perfect white-beam polarizers, antireflective conducting solar cells, and beyond.