The interest in solar ultraviolet (UV) radiation from the scientific community and the general population has risen significantly in recent years because of the link between increased UV levels at the Earth's surface and depletion of ozone in the stratosphere. As a consequence of recent research, UV radiation climatologies have been developed, and effects of some atmospheric constituents (such as ozone or aerosols) have been studied broadly. Correspondingly, there are well-established relationships between, for example, total ozone column and UV radiation levels at the Earth's surface. Effects of clouds, however, are not so well described, given the intrinsic difficulties in properly describing cloud characteristics. Nevertheless, the effect of clouds cannot be neglected, and the variability that clouds induce on UV radiation is particularly significant when short timescales are involved. In this review we show, summarize, and compare several works that deal with the effect of clouds on UV radiation. Specifically, works reviewed here approach the issue from the empirical point of view: Some relationship between measured UV radiation in cloudy conditions and cloud-related information is given in each work. Basically, there are two groups of methods: techniques that are based on observations of cloudiness (either from human observers or by using devices such as sky cameras) and techniques that use measurements of broadband solar radiation as a surrogate for cloud observations. Some techniques combine both types of information. Comparison of results from different works is addressed through using the cloud modification factor (CMF) defined as the ratio between measured UV radiation in a cloudy sky and calculated radiation for a cloudless sky. Typical CMF values for overcast skies range from 0.3 to 0.7, depending both on cloud type and characteristics. Despite this large dispersion of values corresponding to the same cloud cover, it is clear that the cloud effect on UV radiation is 15--45% lower than the cloud effect on total solar radiation. The cloud effect is usually a reducing effect, but a significant number of works report an enhancement effect (that is increased UV radiation levels at the surface) due to the presence of clouds. The review concludes with some recommendations for future studies aimed to further analyze the cloud effects on UV radiation.