Interaction of Interstellar Matter with Cosmic Microwave Background Radiation : One Possible Clue to Anisotropy of the CMBR and to Time's Arrow of the Universe T K Rai Dastidar$^\iacs$ $^\iacs$ Department of Materials Science, Indian Association for the Cultivation of Science, Calcutta 700032, INDIA. The existence of a non-thermodynamic arrow of time at the atomic level was demonstrated in a recent paper (Mod.Phys.Lett. A13, 1265 (1998)). Here we show that the CMBR is capable of making every particle of the universe follow this arrow of time, and that interaction of the CMBR with interstellar matter provides a possible clue to the observed anisotropies in the CMBR.