Oversight is conducted primarily by the Gang of 8.
The Gang of Eight is a colloquial term for a set of eight leaders within the United States Congress who are briefed on classified intelligence matters by the executive branch. Specifically, the Gang of Eight includes the leaders of each of the two parties from both the Senate and House of Representatives, and the chairs and ranking minority members of both the Senate Committee and House Committee for intelligence as set forth by 50 U.S.C. § 3093(c)(2).
Under normal conditions, the President of the United States is required by Title 50 U.S.C. § 3091(a)(1) to "ensure that the congressional intelligence committees are kept fully and currently informed of the intelligence activities of the United States, including any significant anticipated intelligence activity as required by [the] title." However, under "extraordinary circumstances", when the President thinks "it is essential to limit access" to information about a covert action, 50 U.S.C. § 3093(c)(2) allows the President to limit reporting to the Gang of Eight.
However, there are a host of legislative (hearings, authorization) and executive branch (Inspector Generals, etc.) processes or organizational authorities that are detailed here
The judiciary also has a role through the Foreign Intelligence Surveillance Court, which oversees and decisions warrant applications to spy on foreign agents operating within the US.