Searches for pair production of scalar leptoquarks performed by the ATLAS experiment at Large Hadron Collider are presented. The full Run-2 dataset of $pp$ collisions recorded with the ATLAS detector is exploited, corresponding to an integrated luminosity of 139 fb$^{-1}$. Three classes of final states are considered: an oppositely-charged same-flavour light-lepton pair (either two electrons or two muons) and two jets to search for leptoquarks decaying into a jet and a light-lepton; light-leptons and at least one hadronically decaying $\tau$-lepton, which is optimised for a search for leptoquarks decaying into a top-quark and a $\tau$-lepton; an oppositely-charged same-flavour light-lepton pair and two large-radius jets as from decay products of boosted top quarks for leptoquarks decaying into a top-quark and a light-lepton. Various leptoquark models resulting in different choices of quark- and lepton-generations in the decays are tested. No significant excesses above the Standard Model expectations are observed in any of the considered final states, and 95% CL upper limits are set on the production cross-section as a function of the leptoquark mass and the branching fraction, where the branching fraction is considered between two decay modes with charged-leptons and neutrinos for given configurations of quark- and lepton-generations. Under the assumption of the branching fraction into a charged lepton and a quark of 100%, leptoquarks with masses below 1.4 -1.8 TeV are excluded, depending on the leptoquark models.