ALICE is the experiment specifically designed for the study of the Quark-Gluon Plasma in heavy-ion collisions at the CERN LHC. The forward Muon Spectrometer of ALICE is devoted to the study of muon production in the pseudo-rapidity interval between 2.5 and 4. During Run 1 and 2, the spectrometer produced an impressive amount of results concerning the Quarkonia (J/$\Psi$ and $\Upsilon$ families), Heavy flavor and low mass vector mesons. The present spectrometer is composed of two systems: the muon tracking based on 10 planes of Cathode Pad Chambers (CPC) and the muon trigger based on 4 planes of Resistive Plate Chambers (RPC). For Run 3 and 4, an ambitious upgrade program is ongoing for both systems, consisting of the replacement of the front-end and the back-end readout electronics in order to cope with an increased rate by an order of magnitude in Pb-Pb collisions compared to run 2. An additional detector, the Muon Forward Tracker (MFT), will be installed in front of the spectrometer absorber. It consists of planes of high resolution CMOS pixel sensors and it will bring precise vertexing capabilities to the Muon Spectrometer. The technical challenges and construction status related to the muon spectrometer upgrade and the addition of the MFT will be presented.