An Arbitrary Lagrangian-Eulerian formulation has been posed to solve the challenging problem of the three-dimensional Taylor bubble, within a Conservative Level Set (CLS) framework. By employing a domain optimization method (i.e. the moving mesh method), smaller domains can be used to simulate rising bubbles, thus saving computational resources. As the method requires the use of open boundaries, a careful treatment of both inflow and outflow boundary conditions has been carried out. The coupled CLS - moving mesh method has been verified by means of extensive numerical tests. The challenging problem of the full three-dimensional Taylor bubble has then been thoroughly addressed, providing a detailed description of its features. The study also includes a sensitivity analyses with respect to the initial shape of the bubble, the initial volume of the bubble, the flow regime and the inclination of the channel. (C) 2017 Elsevier Ltd. All rights reserved.