A system for automatically inspecting the roundness of containers, comprising: an apparatus for rotating a container about an axis of the container, while the container is being inspected at an inspection station; a first ultrasonic proximity sensor positioned at a fixed location with respect to said inspection station for sensing the distance between said first proximity sensor and the container being inspected at the inspection station while the container is rotated; said first ultrasonic proximity sensor being oriented orthogonally with respect to the axis of the container; and a first processor that processes the distance sensed by said first proximity sensor and provides a first output signal that is indicative of whether or not the distance sensed by said first proximity sensor is within a first predetermined range of distances; wherein: the container, when being inspected at the inspection station, is located a nominal distance X from said first proximity sensor, and a tolerance Y from perfect roundness is acceptable for the container, and said first predetermined range of distances comprises: the range of distances between X-Y and X+Y; the first processor provides a first output signal when the distance between said first proximity sensor and the container is between X-Y and X+Y, and a different second output signal when the distance between said first proximity sensor and the container is either less than X-Y or more than X+Y; and said first processor has a hysteresis function whereby any occurrence of a sensed distance between said first proximity sensor and the container being inspected that is either less than X-Y or more than X+Y must occur for a minimum duration of container rotation before said second output signal is provided by said first processor. an apparatus for rotating a container about an axis of the container, while the container is being inspected at an inspection station; a first ultrasonic proximity sensor positioned at a fixed location with respect to said inspection station for sensing the distance between said first proximity sensor and the container being inspected at the inspection station while the container is rotated; said first ultrasonic proximity sensor being oriented orthogonally with respect to the axis of the container; and a first processor that processes the distance sensed by said first proximity sensor and provides a first output signal that is indicative of whether or not the distance sensed by said first proximity sensor is within a first predetermined range of distances; wherein: the container, when being inspected at the inspection station, is located a nominal distance X from said first proximity sensor, and a tolerance Y from perfect roundness is acceptable for the container, and said first predetermined range of distances comprises: the range of distances between X-Y and X+Y; the first processor provides a first output signal when the distance between said first proximity sensor and the container is between X-Y and X+Y, and a different second output signal when the distance between said first proximity sensor and the container is either less than X-Y or more than X+Y; and said first processor has a hysteresis function whereby any occurrence of a sensed distance between said first proximity sensor and the container being inspected that is either less than X-Y or more than X+Y must occur for a minimum duration of container rotation before said second output signal is provided by said first processor. the container, when being inspected at the inspection station, is located a nominal distance X from said first proximity sensor, and a tolerance Y from perfect roundness is acceptable for the container, and said first predetermined range of distances comprises: the range of distances between X-Y and X+Y; the first processor provides a first output signal when the distance between said first proximity sensor and the container is between X-Y and X+Y, and a different second output signal when the distance between said first proximity sensor and the container is either less than X-Y or more than X+Y; and said first processor has a hysteresis function whereby any occurrence of a sensed distance between said first proximity sensor and the container being inspected that is either less than X-Y or more than X+Y must occur for a minimum duration of container rotation before said second output signal is provided by said first processor. the range of distances between X-Y and X+Y; the first processor provides a first output signal when the distance between said first proximity sensor and the container is between X-Y and X+Y, and a different second output signal when the distance between said first proximity sensor and the container is either less than X-Y or more than X+Y; and said first processor has a hysteresis function whereby any occurrence of a sensed distance between said first proximity sensor and the container being inspected that is either less than X-Y or more than X+Y must occur for a minimum duration of container rotation before said second output signal is provided by said first processor.