Heat oil in a large skillet over medium-high heat. When oil shimmers, add garlic and ginger and saute until fragrant, about 1 minute.
Stir curry paste into garlic and ginger and continue to cook 1 minute more. Add coconut milk, salt, and pepper to skillet and bring to a boil.
Add mussels to skillet, cover, and cook just until mussels open, 2-3 minutes. Transfer mussels to a large serving bowl with a slotted spoon. Pour coconut curry sauce over mussels, garnish with cilantro and lime and serve immediately.