This　paper　addresses　the　consensus　problem　for　complex　dynamical　networks　with　each　node　being　a　Lure　system　whose　nonlinearity　satisfies　a　sector　condition.　Communication　time-delay　and　Markovian　switching　topologies　are　also　considered　in　the　dynamical　network　system.　The　switching　for　topologies　is　determined　by　a　Markov　chain,　each　topology　corresponding　to　a　state　of　the　Markov　chain.　The　partial　stability　theory,　Lyapunov-Krasovskii　function　and　Schur　complement　formula　are　applied　to　get　the　novel　criterion,　which　have　the　forms　of　bilinear　matrix　inequalities.　Moreover,　the　method　of　designing　a　static　output　controller　is　provided.　Finally,　numerical　examples　are　presented　to　illustrate　the　efficiency　and　applicability　of　the　proposed　methods.　©　2017　IEEE.