In this work we study some tests for the homogeneity between two independent samples of functional data. The null hypothesis of ‘homogeneity’ here means that the latent stochastic processes which generated the two samples are actually identically distributed. Because it is generally not possible to define a probability density for functional data, it seems natural to opt for nonparametric procedures in this setting. Making use of recent developments on functional depths, we adapt some Kolmogorov-Smirnov- and Cramer-von-Mises-type of criteria to the functional context. Exact p-values for the test can be obtained via permutations, or in case of too large samples, a bootstrap algorithm is easily implemented. A simulation study illustrates how powerful the devised methodology is, and some real data examples are analysed. Finally, the extension to the case of more than two samples is discussed.