A tubular handling apparatus including a tubular member running tool adapted to provide load-bearing, and preferably torquing, capacity upon the gripping of a tubular is provided. The running tool includes a slotted member having a plurality of elongated slots, a recessed member associated with the slotted member and having a plurality of recesses, and a plurality of gripping elements disposed between the slotted member and recessed member. Each such gripping element is adapted to move with an engaged tubular so as to grip the tubular. A tubular member elevator associated to the running tool, as well as related floor slips, are also encompassed. Methods of casing running are also included.