<p>We prove, under various natural hypotheses, that the random multidimensional affine recursion <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X_n=A_nX_{n-1}+B_n\in \mathbb {R}^d, n \ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, is recurrent in the critical case. In particular we cover the cases where the matrices <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(A_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> are similarities, invertible, rank 1 or with non negative coefficients. These results are a consequence of a criterion of recurrence for a large class of affine recursions on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, based on some moment assumptions of the so-called “reverse norm control random variable.</p>