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Let <i>A</i> and <i>B</i> be Borel subsets of the Euclidean <i>n</i>-space with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">dim</mo><mi>A</mi><mo>+</mo><mo form="prefix">dim</mo><mi>B</mi><mo>></mo><mi>n</mi></mrow></semantics></math></inline-formula>. This is a survey on the following question: what can we say about the Hausdorff dimension of the intersections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>∩</mo><mo>(</mo><mi>g</mi><mo>(</mo><mi>B</mi><mo>)</mo><mo>+</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> for generic orthogonal transformations <i>g</i> and translations by <i>z</i>?