In this paper, the problem of estimating the Intrinsic Dimension of a data set is investigated. A fractal-based approach using the Grassberger-Procaccia algorithm is proposed. Since the Grassberger-Procaccia algorithm performs badly on sets of high dimensionality, an empirical procedure, that improves the original algorithm, has been developed. The procedure has been tested on data sets of known dimensionality and on time series of Santa Fe competition.