We give an explicit classification of the possible p-adic Galois representations that are attached to elliptic curves E with CM defined over ℚ (j (E)). More precisely, let K be an imaginary quadratic field, and let 𝒪 K, f be an order in K of conductor f≥ 1. Let E be an elliptic curve with CM by 𝒪 K, f, such that E is defined by a model over ℚ (j (E)). Let p≥ 2 be a prime, let G ℚ (j (E)) be the absolute Galois group of ℚ (j (E)), and let ρ E, p∞: G ℚ (j (E))→ GL(2, ℤ p) be the Galois representation associated to the Galois action on the Tate module T p (E). The goal is then to describe, explicitly, the groups of GL(2, ℤ p) that can occur as images of ρ E, p∞, up to conjugation, for an arbitrary order 𝒪 K, f.