Recommender　systems　that　predict　the　preference　of　users　have　attracted　more　and　more　attention　in　decades.　One　of　the　most　popular　methods　in　this　field　is　collaborative　filtering,　which　employs　explicit　or　implicit　feedback　to　model　the　user–item　connections.　Most　methods　of　collaborative　filtering　are　based　on　matrix　completion　techniques　which　recover　the　missing　values　of　user–item　interaction　matrices.　The　low-rank　assumption　is　a　critical　premise　for　matrix　completion　in　recommender　systems,　which　speculates　that　most　information　in　interaction　matrices　is　redundant.　Based　on　this　assumption,　a　large　number　of　methods　have　been　developed,　including　matrix　factorization　models,　rank　optimization　models,　and　frameworks　based　on　neural　networks.　In　this　paper,　we　first　provide　a　brief　description　of　recommender　systems　based　on　matrix　completion.　Next,　several　classical　and　state-of-the-art　algorithms　related　to　matrix　completion　for　collaborative　filtering　are　introduced,　most　of　which　are　based　on　the　assumption　of　low-rank　property.　Moreover,　the　performance　of　these　algorithms　is　evaluated　and　discussed　by　conducting　substantial　experiments　on　different　real-world　datasets.　Finally,　we　provide　open　research　issues　for　future　exploration　of　matrix　completion　on　recommender　systems.　©　2021,　The　Author(s),　under　exclusive　licence　to　Springer-Verlag　London　Ltd.,　part　of　Springer　Nature.