In an 8 or 10 quart pot combine apples and cider or juice. Bring to boiling; reduce heat. Cover simmer for 30 minutes, stirring occasionally. Then puree in a food processor or blender. Return to pot. Stir in sugar, cinnamon, cloves, and allspice. Bring to boiling; reduce heat. Cook uncovered, over very low heat 11/2 hours or till very thick, stirring often.
Ladle into hot, sterilized half-pint jars, leaving a 1/4-inch headspace. Wipe jars rims; place lids on jars and screw on the jar rings. Process in boiling-water canner 5 minutes (start timing when water begins to boil). Remove jars and cool on racks. Makes 5-6 half-pints.