As the current solar cycle 25 progresses and moves towards solar maxima, solar activity is increasing and extreme space weather events are taking place. Two severe geomagnetic storms accompanied by two large Forbush decreases in galactic cosmic ray intensity were recorded in March and May, 2024. More precisely, on 24 March 2024, a G4 (according to the NOAA Space Weather Scale for Geomagnetic Storms) geomagnetic storm was registered, with the corresponding geomagnetic indices Kp and Dst equal to 8 and −130 nT, respectively. On the same day, the majority of ground-based neutron monitor stations recorded an unusual Forbush decrease. This event stands out from a typical Forbush decrease because of its high amplitude decrease phase and rapid recovery phase, i.e., 15% decrease and an extremely rapid recovery of 10% within 1.5 h, as recorded at the Oulu neutron monitor station. Furthermore, on 10–13 May 2024, an unusual G5 geomagnetic storm (geomagnetic indices Kp = 9 and Dst = −412 nT) was registered (the last G5 storm had been observed in 2003). In addition, the polar neutron monitor stations recorded a Ground Level Enhancement (GLE74) during the recovery phase of a large Forbush decrease of 15%, which started on 10 May 2024. In this study, a detailed analysis of these two severe events in regard to the accompanying solar activity, interplanetary conditions and solar energetic particle events is provided. Moreover, the results of the NKUA “GLE Alert++ system”, the NKUA/IZMIRAN “FD Precursory Signals” method and the NKUA “ap Prediction tool” concerning these events are presented.