A hydraulically driven diaphragm pumping machine, in particular for pumping difficult-to-pump materials, the pump comprising at least one pump cylinder that has a first end with a first inlet and outlet for fluid to be pumped and a second end with a second inlet and outlet for hydraulic fluid, the inlets and outlets being associated with respective valves, a separator located inside and movable to-and-fro along the pump cylinder, the movable separator having a first side facing the first end of the cylinder and a second side facing the second end of the cylinder, wherein: - the movable separator is connected to the inside of the first end of the cylinder by a first flexible diaphragm in the form of a concertina-like, bellows that is expandable and contractable inside the cylinder along the length direction of the cylinder as the movable separator moves to-and-fro along the cylinder, the first side of the movable separator delimiting a first chamber (c) inside the expandable and contractable flexible diaphragm for containing a variable volume of pumped fluid in communication with the first inlet and outlet; - the movable separator is connected to the inside of the second end of the cylinder by a second flexible diaphragm in the form of a concertinalike bellows that is contractable and expandable along the length direction of the cylinder in correspondence with expansion and contraction of the first flexible diaphragm, the second side of the movable separator delimiting a second chamber (b) inside the second expandable and contractable diaphragm for containing a variable volume of hydraulic fluid in communication with the second inlet and outlet; and - an annular space (a) is defined between the outside of the first and second diaphragms (4,10) and the inner wall of the pump cylinder, which annular space (a) in use contains a fluid that is the same as said hydraulic fluid or has similar hydraulic characteristics, characterized in that: the movable separator is in the form of a plunger that is slidably mounted in the middle part of the inside of the cylinder between the first and second bellows-like diaphragms (4,10), one end of the plunger being connected to the first bellows-like diaphragm arid the other end of the plunger being connected to the second bellows-like diaphragm to define respective first and second annular spaces (a), namely a first annular space (a) between the outside of the first bellows-like diaphragm and the inner wall of the pump cylinder and a second annular space (a) between the outside of the second bellows-like diaphragm and the inner wall of the pump cylinder, wherein the first and second annular spaces (a) are independent of one another and the pressure of fluid in the first annular space (a) is independent of the pressure of fluid in the second annular space (a). - the movable separator is connected to the inside of the first end of the cylinder by a first flexible diaphragm in the form of a concertina-like, bellows that is expandable and contractable inside the cylinder along the length direction of the cylinder as the movable separator moves to-and-fro along the cylinder, the first side of the movable separator delimiting a first chamber (c) inside the expandable and contractable flexible diaphragm for containing a variable volume of pumped fluid in communication with the first inlet and outlet; - the movable separator is connected to the inside of the second end of the cylinder by a second flexible diaphragm in the form of a concertinalike bellows that is contractable and expandable along the length direction of the cylinder in correspondence with expansion and contraction of the first flexible diaphragm, the second side of the movable separator delimiting a second chamber (b) inside the second expandable and contractable diaphragm for containing a variable volume of hydraulic fluid in communication with the second inlet and outlet; and - an annular space (a) is defined between the outside of the first and second diaphragms (4,10) and the inner wall of the pump cylinder, which annular space (a) in use contains a fluid that is the same as said hydraulic fluid or has similar hydraulic characteristics, the movable separator is in the form of a plunger that is slidably mounted in the middle part of the inside of the cylinder between the first and second bellows-like diaphragms (4,10), one end of the plunger being connected to the first bellows-like diaphragm arid the other end of the plunger being connected to the second bellows-like diaphragm to define respective first and second annular spaces (a), namely a first annular space (a) between the outside of the first bellows-like diaphragm and the inner wall of the pump cylinder and a second annular space (a) between the outside of the second bellows-like diaphragm and the inner wall of the pump cylinder, wherein the first and second annular spaces (a) are independent of one another and the pressure of fluid in the first annular space (a) is independent of the pressure of fluid in the second annular space (a).