Abstract: To every simple graph one associates its edge ideal which is generated by quadratic squarefree monomials corresponding to edges of the graph. In this talk, we study the Alexander dual of edge ideals, which is called the cover ideal. The reason for this naming is that the cover ideal is minimally generated by squarefree monomials corresponding to the minimal vertex covers of the given graph. We review the recent results about the symbolic powers of cover ideals. In particular, we characterize all graphs with the property that every symbolic power of its cover ideal has a linear resolution. Also, we determine an upper bound for the regularity of symbolic powers of certain classes of graphs including bipartite graphs, unmixed graphs and claw-free graphs. Moreover, we study the asymptotic behavior of depth of symbolic powers of cover ideals