Leptoquarks (LQ), introduced in several Beyond the Standard Model (BSM) theories, can provide a possible explanation of potential violations of lepton flavour universality in measurements of B-meson decays (``B-anomalies''), and of the $g-2$ anomaly measured at Fermilab. Recent results from the ATLAS Collaboration on scalar and vector leptoquark pair production and decay in third generation quarks and first- or second-generation leptons (LQ$_{\rm mix}$) are presented. The analyses have been performed on a dataset corresponding to 139 fb$^{-1}$, recorded at the LHC at a centre of mass energy of $\sqrt{s} = $ 13 TeV. No evidence of new physics has been found, and lower limits on the mixed-generation leptoquark LQ$_{\rm mix}$ mass have been set, both scalar and vector, up/down type, and assuming different coupling scenarios for vector leptoquarks: Yang-Mills (vLQ$_{\rm YM}$) or minimal coupling (vLQ$_{\rm min}$).