A cellular composition comprising a pharmaceutically acceptable carrier and cells derived from umbilicus derived substantially free human cord blood, for use in inhibiting an adverse immune response in a transplant recipient is coincident histocompatibility for giving transplant, wherein said cells are derived from tissue substantially free umbilical cord blood, are capable of self-renewal and expansion in culture, have the potential to differentiate into cells of at least one neural phenotype, in which the cells require L valine for growth, produce CD45 and CD117 and can grow in at least about 5% oxygen, and wherein the cells comprise the following characteristics: a) potential for at least 40 doublings in culture b) bonding and expansion in a culture vessel coated or uncoated fabric, in which a culture vessel coated fabric comprises a coating of gelatin, laminin, collagen, polyornithine, vitronectin or fibronectin c) production of vimentin and actin smooth muscle alpha d) production of CD10, CD13, CD44, CD73 and CD90 e) expression of a gene, which, relative to a human cell that is a fibroblast, a mesenchymal stem cell, or a cell of bone marrow from the iliac crest, is increased for a gene encoding interleukin 8 and reticulon 1.Una composición celular que comprende un vehículo farmacéuticamente aceptable y las células derivadas de ombligo derivadas de cordón umbilical humano sustancialmente libre de sangre, para su uso en la inhibición de una respuesta inmune adversa en un receptor de trasplante que es de histocompatibilidad coincidente para el donante del trasplante, en que dichas células se derivan de tejido del cordón umbilical sustancialmente libre de sangre, son capaces de auto-renovación y expansión en cultivo, tienen el potencial de diferenciarse en células de al menos un fenotipo neural, en el que las células requieren L-valina para el crecimiento, no producen CD45 y CD 117 y pueden crecer en al menos aproximadamente 5% de oxígeno, y en el que las células c
