Abstract Kwapień’s theorem asserts that every continuous linear operator from $$\ell _{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> to $$\ell _{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> is absolutely $$\left( r,1\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> </mml:math> -summing for $$1/r=1-\left| 1/p-1/2\right| .$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mfenced> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> When $$p=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> it recovers the famous Grothendieck’s theorem. In this paper we investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapień’s and Grothendieck’s results that encompasses the cases of multiple summing and absolutely summing multilinear operators.