An unstructured conservative level-set method for two-phase flow with variable surface tension is introduced. Surface tension is a function of temperature or surfactant concentration on the interface. Consequently, the called Marangoni stresses induced by temperature gradients or surfactant concentration gradients on the interface lead to a coupling of momentum transport equation with thermal energy transport equation or interface surfactant transport equation. The finite-volume method discretizes transport equations on 3D collocated unstructured meshes. The unstructured conservative level-set method is employed for interface capturing, whereas the multiple marker approach avoids the numerical coalescence of fluid particles. The fractional-step projection method solves the pressure-velocity coupling. Unstructured flux-limiters are proposed to discretize the convective term of transport equations. A central difference scheme discretizes diffusive terms. Gradients are evaluated by the weighted least-squares method. Verifications and validations are reported