For a set D⊂Z_n, the distance graph P_n(D) has Z_n as its vertex set and the edges are between vertices i and j with |i-j|∈D. The circulant graph C_n(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.