The cutwidth of a graph G is defined to be the smallest integer k such that the vertices of G can be arranged in a linear layout [v(1),...,v(n)] in such a way that for every i=1,...,n-1, there are at most k edges with the one endpoint in {v(1),...,v(i)} and the other in {v(i+1),...,v(n)}. In this paper we show how to construct, for any constant k, a linear time algorithm that for any input graph G, answers whether G has cutwidth at most k and, in the case of a positive answer, outputs the corresponding linear layout.