Solidification structures of REBCO(RE=Y, Nd) fabricated by the infiltration-growth method were experimentally and analytically studied. Ba-Cu-O liquid with a small amount RE was infiltrated into pre-sintered porous Y211(Y2BaCuO5) or Nd422(Nd4Ba2Cu2O10) solid, and then cooled under the peritectic temperature to grow 123 crystal under a temperature gradient. The effects of Ag-addition on the structure of REBCO were studied. The critical transition conditions from columnar to equiaxed structures were investigated for 123-macrostructures in unidirectionally solidified REBCO. The undercooling(Delta T) for crystal growth and that for nucleation were evaluated analytically to obtain the transition condition by using the growth rate(R)-Delta T relation and the cooling rate(Vc)-Delta T relation for 123 crystals. The calculated results for the critical transition condition from columnar to equiaxed-structure showed good agreement with the experimental results. The formation process of microstructure of REBCO was also studied by two-dimensional numerical simulation of faceted growth.