Contaminants on large astronomical reflecting surfaces (hereafter "mirrors") can significantly degrade their reflectivity, IR emissivity, and light-scattering properties. We will show data that the emissivity and scattering can degrade appreciably after just a few days or weeks. A safe, effective, and inexpensive cleaning technique, preferably one that can be used in situ, is especially important for 4-to 8-m class mirrors. Two cleaning methods CO 2 snow and pulsed ultraviolet (UV) lasers, offer the potential to satisfy these needs. Our primary purpose is to compare the two methods, and highlight their advantages and problems. Also, since the UV-laser cleaning technique is a new one, we describe how it works and how it may be implemented. We found that UV-laser cleaning removes contaminants that standard CO 2-snowcleaning techniques do not remove as well or not at all. After 2-4 weeks of exposure, Al-coated-mirror samples placed under the mirror covers of the UKIRT telescope on Mauna Kea and the 4-m telescope on Kitt Peak were cleaned about twice as effectively by the UV laser than the CO 2 snow, based upon the amount of contamination left on the mirrors. For long exposure times, the laser cleaning also restores the thermal emissivity better than the C02 snow. While a CO 2-snow delivery system can be less expensive than a UV-Iaser-cleaning system, the operation costs for cleaning a large-diameter mirror with CO 2 snow can be substantially greater than the costs to run the laser. Unlike C02 snow, laser cleaning can be applied no matter what the local humidity. The ease of both methods facilitates frequent use; however, the cost of the CO 2 may eventually limit its usage.