Extremely smooth single crystal YBa2Cu3O7-delta "thin" films, 1-3 mum thick, have been successfully grown on YBCO-seeded MgO substrates by liquid phase epitaxy. The morphology study on the as-grown samples has revealed a step-flow growth mechanism, with each step height of about 1.1 nm, i.e. the c-axis lattice constant of YBCO. The mean surface roughness in a large 25 mum x 25 mum area is similar to0.76nm, determined by an atomic force microscope. After annealing in pure oxygen, the similar to2 mum thick films exhibit high-quality high- T-c superconductivity with zero resistance transition temperature T-C0 approximate to 91 K and critical current density J(C) = 4.74 x 10(4) A/cm(2) (transport measurement with 1 muV/cm criterion) at 77 K. (C) 2002 Elsevier Science BY. All rights reserved.