Human-pluripotent-stem-cell-derived kidney cells (hPSC-KCs) have important potential for disease modelling and regeneration. Whether the hPSC-KCs can reconstitute tissue-specific phenotypes is currently unknown. Here we show that hPSC-KCs self-organize into kidney organoids that functionally recapitulate tissue-specific epithelial physiology, including disease phenotypes after genome editing. In three-dimensional cultures, epiblast-stage hPSCs form spheroids surrounding hollow, amniotic-like cavities. GSK3β inhibition differentiates spheroids into segmented, nephron-like kidney organoids containing cell populations with characteristics of proximal tubules, podocytes and endothelium. Tubules accumulate dextran and methotrexate transport cargoes, and express kidney injury molecule-1 after nephrotoxic chemical injury. CRISPR/Cas9 knockout of podocalyxin causes junctional organization defects in podocyte-like cells. Knockout of the polycystic kidney disease genes PKD1 or PKD2 induces cyst formation from kidney tubules. All of these functional phenotypes are distinct from effects in epiblast spheroids, indicating that they are tissue specific. Our findings establish a reproducible, versatile three-dimensional framework for human epithelial disease modelling and regenerative medicine applications.