A vulcanizable elastomer composition is prepared by blending 100 parts by weight of a copolymer containing(A) 50-85% by mole of ethylene unit,(B) 10-45% by mole of an alkyl acrylate unit having the formula (I), ##STR1## wherein R.sub.1 is an alkyl having 1-8 carbon atoms, (C) 2-35% by mole of an alkoxyalkyl acrylate unit having the formula (II), ##STR2## wherein R.sub.2 is hydrogen or methyl, R.sub.3 is an alkylene having 1-4 carbon atoms, and R.sup.4 is an alkyl or alkoxyalkyl, each having 1-4 carbon atoms, and(D) up to 5% by mole of a glycidyl ester unit having the formula (III), ##STR3## wherein R.sup.5 is hydrogen or methyl, and 0.1-10 parts by weight of at least one vulcanizing agent.The vulcanizable composition can give vulcanizates of well-balanced properties having an improved low-temperature resistance and retaining excellent oil resistance.