Any general singular fiber of a proper holomorphic Lagrangian fibration is foliated by natural 1-cycles called the characteristic 1-cycles. Except one case where the cycle will be of infinite chain, it turns out that they are exactly of the same form as singular fibers of relatively minimal elliptic surfaces. In the talk, after introducing the notion of the characteristic 1-cycle, I would like to explain our complete classification of general singular fibers counted with multiplicity, in terms of the characteristic 1-cycles, together with existence.