The present invention features novel methods for the cryopreservation of mammalian cell that combine the advantages of the slow-freezing and vitrification approaches while avoiding their shortcomings. Generally, the methods include the use of a capillary tube made of a thermally conductive wall material and a thin wall such that the ratio of the thermal conductivity of the wall material to the wall thickness is at least 1,000-500,000. The solution is then exposed to temperatures equal to or less than −80° C. and the vitrification solution containing the mammalian cells is cooled at a rate equal to or greater than 30,000-100,000,000° C./minute. The exposure of the capillary tube with a thermally conductive and thin wall allows for vitrification of the solution in the absence of ice formation. Cryoprotectants can also be added to the vitrification solution to further prevent ice formation.