An apparatus for conveying viscous material out of a barrel-like container having a container bottom and a container wall that extends upward from the container bottom has a follower plate for closing off the container, which plate is movable in the direction toward the container bottom, lying against the inner surface of the container wall, which surface faces the container interior, and which plate has a dome that extends upward from an underside intended to lie on the viscous material, which dome opens, in the center, into a material exit opening. The apparatus has a pump for conveying the viscous material through the material exit opening. At least one displacement body projects into the dome through the material exit opening and is connected with the pump for displacement of the material situated in the dome and/or air situated in the dome, and at least one closeable venting opening vents the dome.