A well pipe handling machine for removing a series of stands of pipe from the upper end of a well pipe string and storing the stands in a rack offset to a side of the well axis and/or returning the stands sequentially from the rack for reconnection to the string. The machine includes a support which carries units for holding a pipe in vertical condition and is movable between a position in which the pipe is in alignment with the well axis and a location near the storage rack, and which preferably takes the form of a vertically extending column structure movable horizontally by two synchronized drive mechanisms at the top and bottom of the column structure. The pipe holding units are desirably shiftable upwardly and downwardly relative to the support and also generally horizontally toward and away from the support for placement of the pipe stands in the rack and removal therefrom. The pipe holding units may include two vertically spaced pipe gripping units mounted by a parallelogram mechanism for movement toward and away from the support. The machine also may include apparatus for turning a pipe stand to connect it to or disconnect it from the string, with this apparatus being movable with the support and pipe holding units between the well axis and the storage rack.