One of the probable sources of water and other volatiles, forming ice deposits in the shadowed polar regions of the Moon, is cosmic matter delivered by comets and asteroids, as well as by cosmic dust. In this connection, we estimated the contribution of volatile components to the material received due to micrometeoroid bombardment. From the estimates of the current flux of micrometeoroids to the Earth and the Moon, we determined a probable range for the rate of the cosmic dust accretion to the Moon as 6.19 x 10(-13)to 14.74 x 10(-13)g/m(2)s. Based on the mean value of this rate, the mass of fallen micrometeoroids is equal to the total mass of cosmic matter delivered to the Moon by bodies 3 m to 4 km in size. Taking into consideration that the largest fraction (similar to 90%) of the mass of micrometeoroids is represented by the material that is close to CM- and CI-type carbonaceous chondrites in composition, we estimated the amounts of released water and other components and volatile elements. The obtained values are close to the amount of water introduced by comets.