1. Евге́ний (Yevgéniy) This is the most common and traditional way to spell "Eugene" in Russian. It's used for both male and female names.
2. Юджин (Yugin) This is a more recent transliteration of the name, used for both male and female names. It's based on the English pronunciation of the name.
The choice between the two depends on the specific context. If you're referring to a historical figure or a well-known person, you'll likely use Евге́ний. If you're writing about a contemporary person or a fictional character, you might use Юджин.
Ultimately, both spellings are acceptable and will be understood by most Russian speakers.