A fat-containing composition obtained by converting insects or worms into nutrient streams, such as a fat-containing, an aqueous protein-containing and a solid-containing fraction, including (a) squashing insects or worms thereby obtaining a pulp, the insects or worms reduced in size, (b) heating the pulp to 70-100° C., and (c) subjecting the heated pulp to a physical separation step with the proviso that the method does not contain enzymatic treatment of the pulp. The fat-containing fraction contains at least 80 wt. % insect or worm fat with at least 40 wt. % saturated fats. The aqueous protein fraction can be dried to obtain dried protein material, which contains at least 50 wt. % insect or worm protein-derived matter and at most 25 wt. % insect or worm fat based on dry weight. The protein has a pepsin digestibility of at least 50%. The resulting nutrient streams can be used in food, petfood, feed and pharmaceutical industry.