The edge Mostar index 𝑀𝑜𝑒(𝐺) of a connected graph 𝐺 is defined as 𝑀𝑜𝑒(𝐺)=Σ𝑒=𝑢𝑣∈𝐸(𝐺) |𝑚𝑢(𝑒|𝐺)−𝑚𝑣(𝑒|𝐺)|, where 𝑚𝑢(𝑒|𝐺)and 𝑚𝑣(𝑒|𝐺) are, respectively, the number of edges of 𝐺 lying closer to vertex 𝑢 than to vertex 𝑣 and the number of edges of 𝐺 lying closer to vertex 𝑣 than to vertex 𝑢. In this paper, we determine the extremal values of edge Mostar index of some graphs. We characterize extremal trees, unicyclic graphs and determine the extremal graphs with maximum and second maximum edge Mostar index among cacti with size 𝑚 and 𝑡 cycles. At last, we give some open problems.