A machine for making liquid or semi-liquid food products, comprising: - a frame; - a plurality of first containers containing a liquid or semi-liquid basic mixture and comprising a deformable wall, so that the space occupied by each first container depends on a quantity of basic mixture in the self-same first container; - a rigid second container forming a processing chamber for said basic mixture and provided with a mixer and with thermal treatment means for the basic mixture, operating in conjunction with one another to convert said basic mixture into a liquid or semi-liquid product; - connecting means for connecting the first containers to the second container, configured to allow the operative connection of one of said first containers to the second container and to transfer the basic mixture from said one of said first containers to the second container; - supporting means for said first containers, operatively associated with said frame, said machine being characterised in that it further comprises: - at least one sensor configured to make available a signal representative of the quantity of basic mixture at least in one of the first containers which is operatively connected to the second container; - a control and actuator unit, connected to said sensor for receiving said signal and configured to infer, based on said signal, a condition of absence of mixture in said one of the first containers which is operatively connected to the second container, the machine being characterized in that the connecting means for connecting the first containers to the second container comprise at least a transfer pump configured to allow a transfer of basic mixture from the first containers to the second container, and are configured to allow selective connection of the transfer pump to, and disconnection of said pump from, each of said first containers, said control unit being connected to the connecting means to control the connecting means for establishing the connection of the transfer pump to one of said first containers, which is completely full, after a condition of absence of mixture in the first container which is connected to the transfer pump has been inferred and the machine being further characterized in that said selective connecting means comprise a plurality of pipes connecting the first containers to the transfer pump, and closing means for closing each of said pipes, which can be activated and deactivated by said control unit to allow the connection of a preset first container, which is completely full, to the transfer pump. - a frame; - a plurality of first containers containing a liquid or semi-liquid basic mixture and comprising a deformable wall, so that the space occupied by each first container depends on a quantity of basic mixture in the self-same first container; - a rigid second container forming a processing chamber for said basic mixture and provided with a mixer and with thermal treatment means for the basic mixture, operating in conjunction with one another to convert said basic mixture into a liquid or semi-liquid product; - connecting means for connecting the first containers to the second container, configured to allow the operative connection of one of said first containers to the second container and to transfer the basic mixture from said one of said first containers to the second container; - supporting means for said first containers, operatively associated with said frame, - at least one sensor configured to make available a signal representative of the quantity of basic mixture at least in one of the first containers which is operatively connected to the second container;