An edge-colored graph G is conflict-free connected if, between each pair of distinct vertices of G, there exists a path in G containing a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected. In this paper, we firstly determine all trees T of order n for which cfc(T)=n−t, where t≥1 and n≥2t+2. Secondly, we prove that let G be a graph of order n, then 1≤cfc(G)≤n−1, and characterize the graphs G with cfc(G)=1,n−4,n−3,n−2,n−1, respectively. Finally, we get the Nordhaus–Gaddum-type result for the conflict-free connection number of graphs, and prove that if G and G¯ are connected graphs of order n (n≥4), then 4≤cfc(G)+cfc(G¯)≤n and 4≤cfc(G)⋅cfc(G¯)≤2(n−2), moreover, cfc(G)+cfc(G¯)=n or cfc(G)⋅cfc(G¯)=2(n−2) if and only if one of G and G¯ is a tree with maximum degree n−2 or a path of order 5, and the lower bounds are sharp.