The systems are offset by fixed steps — UK women's sizes typically run four numbers above US, and EU sizes about 30 above US — but those are rules of thumb, not guarantees. Vanity sizing and brand-specific grading mean the same "EU 38" varies between labels.
Conversion charts get you in the right neighbourhood; the brand's own measurements get you the right fit. When shopping European brands like Zara or Mango from the US, always cross-check the centimetre measurements.