A low solubility of yttrium in the Ba-Cu-O melt and a steep liquidus slope near the peritectic temperature Tp lead to a very slow growth rate of YBa2Cu3O7-δ(YBCO or Y123) single crystals and this creates a problem in growth of large single crystals. To solve this problem, increasing the growth rate and extending the growth time are significant. Using the crystal pulling method, we have developed several processes and succeeded in growing large Y123 and Nd1−χBa2+χCu3O7−(NδBCO) single crystals with an edge size over 25mm in the a-b plane and up to 20mm in the c-axis direction. In this article, three methods of increasing growth rate are reviewed. They are (i) employing high oxygen partial pressure, (ii) choosing RE (rare earth) elements with higher solubilities in the Ba-Cu-O solution, and (iii) growing REBCO crystals including several RE elements. Using these methods the growth rate was effectively enhanced from two to five times that of Y123. The critical temperature Tc of 92.7 K was achieved from a Y123 single crystal grown under 1 atm oxygen partial pressure, indicating that Tc is insensitive to the oxygen pressure of the growth atmosphere in the YBCO system. A high Tc of about 95 K for NdBCO superconductors with a sharp transition was obtained by controlling the ratio of Ba to Cu (Ba/Cu) in the liquid, suggesting that the Ba/Cu ratio in the liquid composition has a significant importance in controlling Tc. By partial substitution of Sm at the Y sites up to 30%, Y1−xSmx Ba2Cu3O7−(Y(Sm)BCO) crystals show a Tc of 91±1K and do not display Tc depression.