Let　H　be　a　hypergraph　with　vertex　set　V(H)　and　hyperedge　set　E(H).　We　call　a　vertex　set　R⊆V(H)　a　transversal　if　it　has　a　nonempty　intersection　with　every　hyperedge　of　H.　The　transversal　number,　denoted　by　τ(H),　is　the　minimum　cardinality　of　transversals.　In　2021,　Diao　verified　that　the　upper　bound　of　transversal　number　for　any　connected　3-uniform　hypergraph　H　is　at　most　2m+13,　that　is,　τ(H)⩽2m+13,　where　m　is　the　size　of　H.　Moreover,　they　gave　the　necessary　and　sufficient　conditions　to　reach　the　upper　bound,　namely　τ(H)=2m+13,　if　and　only　if　H　is　a　hypertree　with　a　perfect　matching.　In　this　paper,　we　investigate　the　transversal　number　of　connected　k-uniform　hypergraphs　for　k⩾3.　We　confirm　that　τ(H)⩽(k-1)m+1k　for　any　k-uniform　hypergraph　H　with　size　m.　Furthermore,　we　show　that　τ(H)=(k-1)m+1k　if　and　only　if　H　is　a　hypertree　with　a　perfect　matching,　which　generalizes　the　results　of　Diao.　©　Operations　Research　Society　of　China,　Periodicals　Agency　of　Shanghai　University,　Science　Press,　and　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature　2022.　Springer　Nature　or　its　licensor　holds　exclusive　rights　to　this　article　under　a　publishing　agreement　with　the　author(s)　or　other　rightsholder(s);　author　self-archiving　of　the　accepted　manuscript　version　of　this　article　is　solely　governed　by　the　terms　of　such　publishing　agreement　and　applicable　law.