The Dvoretzky random covering problem is to find the conditions for which almost surely every point on the circle is covered infinitely many times by a sequence of random intervals with decreasing lengths and random initial points (an i.i.d. sequence of random variables uniformly distributed on the circle). It has drawn a lot of interest of many mathematicians for the last decades and the sizes of the random covering sets have been widely studied. The Hausdorff dimensions and hitting probabilities of random covering sets will be given in the talk. The covering setting also was generalized to many different cases, for example, covering the torus with rectangles or open sets, or even just Lebesgue measure sets, or balls with singular distributions, some recent related results will be surveyed.