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In this paper, two new approaches are presented for the solution of electromagnetic inverse scattering problems when amplitude-only data are available. The proposed techniques are based on a customized version, which are the so-called contrast source inversion (CSI) and multiplicative regularized CSI (MRCSI) methods. In the proposed approaches, denoted as the phaseless-data (PD)-CSI and the PD-MRCSI, only the term of the cost functional concerning the mismatch between the measured and estimated field data (i.e., the data equation) has been properly redefined. Moreover, the back-projection algorithm has been modified to provide an initial solution ensuring the rapid convergence of the optimization procedures and avoid the reconstruction of false solutions. A set of representative results concerning numerical as well as experimental tests is reported to show the accuracy of the proposed amplitude-only reconstruction approaches.