Three different permutation test schemes are discussed and compared in the context of the two-sample problem for functional data. One of the procedures was essentially introduced by Lopez-Pintado and Romo (2009), using notions of functional data depth to adapt the ideas originally proposed by Liu and Singh (1993) for multivariate data. Of the new methods introduced here, one is also based on functional data depths, but uses a different way (inspired by Meta-Analysis) to assess the significance of the depth differences. The second new method presented here adapts, to the functional data setting, the k-nearest-neighbors statistic of Schilling (1986). The three methods are compared among them and against the test of Horvath and Kokoszka (2012) in simulated examples and real data. The comparison considers the performance of the statistics in terms of statistical power and in terms of computational cost.