This paper deals with the cyclic job shop problem where the task durations are uncertain and belong to a polyhedral uncertainty set. We formulate the cyclic job shop problem as a two-stage robust optimization model. The cycle time and the execution order of tasks executed on the same machines correspond to the here-and-now decisions and have to be decided before the realization of the uncertainty. The starting times of tasks corresponding to the wait-and-see decisions are delayed and can be adjusted after the uncertain parameters are known. In the last decades, different solution approaches have been developed for two-stage robust optimization problems. Among them, the use of affine policies, row and row-and-column generation algorithms are the most common. In this paper, we propose a branch-andbound algorithm to tackle the robust cyclic job shop problem with cycle time minimization. The algorithm uses, at each node of the search tree, a robust version of the Howard's algorithm to derive a lower bound on the optimal cycle time. Moreover, we design a row generation algorithm and a column-and-row generation algorithm and compare it to the branchand-bound method. Finally, encouraging preliminary results on numerical experiments performed on randomly generated instances are presented.