If　for　any　two　vertices　v(1)　and　v(2)　of　digraph　D,　D　admits　a　spanning　(v(1),　v(2))-dipath　or　a　spanning　(v(2),　v(1))-dipath,　then　D　is　weakly　Hamiltonian-connected;　and　if　there　are　both　a　spanning　(v(1),　v(2))-dipath　and　a　spanning　(v(2),　v(1))-dipath,　then　D　is　strongly　Hamiltonian-connected.　Thomassen　in　[J.　of　Combinatorial　Theory,　Series　B,　28,　(1980)　142-163]　discovered　a　collection　T-0　of　two　digraphs　and　used　it　to　characterize　weakly　Hamiltonian-connected　tournaments,　and　proved　that　every　4-connected　tournament　is　strongly　Hamiltonian-connected.　For　any　two　vertices　v(1)　and　v(2)　of　digraph　D,　D　is　weakly　trail-connected　if　D　admits　a　spanning　(v(1),　v(2))-ditrail　or　a　spanning　(v(2),　v(1))-ditrail,　and　D　is　strongly　trail-connected　if　there　are　both　a　spanning　(v(1),　v(2))-ditrail　and　a　spanning　(v(2),　v(1))-ditrail.　We　have　determined　a　family　T　of　tournaments　and　prove　the　following.　(i)　A　tournament　D　is　weakly　trail-connected　if　and　only　if　D　is　strong.　(ii)　A　strong　tournament　D　is　strongly　trail-connected　if　and　only　if　D　is　not　a　member　in　T.　(iii)　Every　tournament　with　arc-strong　connectivity　at　least　2　is　strongly　trail-connected.　(C)　2020　Elsevier　Inc.　All　rights　reserved.