We　report　on　an　investigation　of　state-of-the-art　flux-grown　multiferroic　bismuth　ferrite　(BiFeO3;　BFO)　single　crystals　by　transmission　electron　microscopy　and　electron　diffraction.　The　crystals　were　pre-characterized　by　piezoresponse　force　microscopy,　electrical　resistance　and　superconducting　quantum　interference　device　magnetization　measurements.　The　structurally　highly　perfect　crystals　show　a　ferroelectric　stripe　domain　structure　characterized　by　a　domain　width　of　55　nm.　Inside　these　domains　an　additional　contiguous　nanodomain　substructure　occurs,　consisting　of　180　degrees　related　domains,　giving　rise　to　satellite　reflections　at　g　{1/2　1/2　1/2}-type　positions　along　＜　110　＞　directions　in　the　electron　diffraction　pattern　corresponding　to　a　characteristic　length　in　real　space　of　15.5　nm.　Furthermore,　we　present　the　first　atomic-resolution　study　on　the　short-range　order　by　aberration-corrected　transmission　electron　microscopy　in　which　all　atoms　including　oxygen　are　imaged　directly.　By　measuring　the　-Fe-O-Fe-　atom　topology,　bond　angles　and　atomic　distances　we　derive　the　electrical　dipole　moment　as　well　as　the　magnitude　of　the　magnetic　moment　on　the　unit-cell　level.　The　results　evidence　substantial　atomic-　to　nano-scale　disorder.　Both　the　nanodomain　substructure　as　well　as　the　disorder　should　affect　the　subtle　magnetoelectric　interactions　in　this　material　and　thereby　impede　the　formation　of　long-range　cycloidal　spin　ordering　which　up　to　now　was　considered　an　intrinsic　feature　of　the　magnetic　properties　of　BiFeO3　single　crystals.　By　Monte　Carlo　simulation　on　the　basis　of　a　state-of-the-art　effective　Hamiltonian　we　scrutinize　certain　aspects　of　the　phase　formation　behavior　in　the　BFO　system　forming　the　background　of　single-crystal　growth.　This　study　reveals　a　very　sluggish　phase　evolution　behavior,　which　should　make　it　invariably　difficult　to　obtain　structurally　fully　equilibrated　single　crystals.　(C)2014　Acta　Materialia　Inc.　Published　by　Elsevier　Ltd.　All　rights　reserved.