The present invention provides olefin polymer compositions that provide an easy-peelable sealant showing low temperature dependence of heat-sealing strength, excellent low-temperature sealability, sealing properties and appearance of the peeling section when used with polypropylene. In particular, the present invention includes as a composition having excellent heat and oil resistance a composition comprising in a particular proportion a propylene-based polymer (A), ethylene-based polymer (B) having a density of not less than 900 kg/m.sup.3, a non-crystalline or low-crystalline ethylene/.alpha.-olefin copolymer (C) and a tackifier resin (D), and as a composition having excellent oil resistance a composition comprising in a particular proportion an ethylene/unsaturated ester copolymer (E), a propylene-based polymer (A), a non-crystalline or low-crystalline ethylene/.alpha.-olefin copolymer (C) and a tackifier resin (D).