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In this systematic review, the authors give a survey on the recent developments of both the John–Nirenberg space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><msub><mi>N</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mi>J</mi><msub><mi>N</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>J</mi><msub><mi>N</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula> or a given cube <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mn>0</mn></msub><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> with finite side length. In addition, some related open questions are also presented.