<p>We study the positivity and asymptotic behaviour of nonnegative solutions of a general nonlocal fast diffusion equation, <Equation ID="Equ42"> <EquationSource Format="TEX">\(\begin{aligned} \partial _t u + \mathcal {L}\varphi (u) = 0, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>+</mo> <mi mathvariant="script">L</mi> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and the interplay between these two properties. Here <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {L}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> is a stable-like operator and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is a singular nonlinearity. We start by analysing positivity by means of a weak Harnack inequality satisfied by a related elliptic (nonlocal) equation. Then we use this positivity to establish the asymptotic behaviour: under certain hypotheses on the nonlocal operator and nonlinearity, our solutions behave asymptotically as the Barenblatt solution of the standard fractional fast diffusion equation. The main difficulty stems from the generality of the operator, which does not allow the use of the methods that were available for the fractional Laplacian. Our results are new even in the case where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is a power.</p>