MPEJ Volume 9, No. 4, 13 pp. Received: Sep 16, 2002. Revised: Sep 10, 2003. Accepted: Sep 18, 2003. V. Sirvent Hilbert's Space Filling Curves and Geodesic Laminations ABSTRACT: We present a modified version of the classical Hilbert's space filling curve and we associate to this curve a geodesic lamination on the disk together with a transversal measure. The lamination helps us to understand how the points of the interval are mapped to the square. We generalize this construction to space filling curves from the interval to the regular $n$-gon. http://www.maia.ub.es/mpej/Vol/9/4.ps http://www.ma.utexas.edu/mpej/Vol/9/4.ps http://mpej.unige.ch/mpej/Vol/9/4.ps