This paper presents a study of holographic superconductors, which are strongly coupled field theories that undergo a superconducting phase transition below a critical temperature. These systems have a gravity dual in the sense of the AdS/CFT correspondence. The authors analyze the behavior of these systems by coupling anti-de Sitter (AdS) gravity to a Maxwell field and a charged scalar field. They consider all values of the charge of the scalar field and find that superconductivity persists even for arbitrarily small charges. In the large charge limit, the backreaction of the scalar field on the spacetime metric is negligible, but for small charges, backreaction becomes important and leads to a new type of black hole instability. The authors also study the effects of a perpendicular magnetic field and derive the London equation and magnetic penetration depth. They show that these holographic superconductors are Type II, meaning they transition from a normal phase to a superconducting phase as the magnetic field is reduced. The authors also compute the conductivities of the system, including electric, thermal, and thermoelectric conductivities. They find that the conductivity exhibits a Drude peak at zero frequency, which is a signature of translationally invariant systems. The authors also discuss the connection between superconductivity and conformal field theories, and show that the AdS/CFT correspondence can be used to describe superconductivity in non-conformal field theories. The paper concludes with a summary of the results and some open questions.This paper presents a study of holographic superconductors, which are strongly coupled field theories that undergo a superconducting phase transition below a critical temperature. These systems have a gravity dual in the sense of the AdS/CFT correspondence. The authors analyze the behavior of these systems by coupling anti-de Sitter (AdS) gravity to a Maxwell field and a charged scalar field. They consider all values of the charge of the scalar field and find that superconductivity persists even for arbitrarily small charges. In the large charge limit, the backreaction of the scalar field on the spacetime metric is negligible, but for small charges, backreaction becomes important and leads to a new type of black hole instability. The authors also study the effects of a perpendicular magnetic field and derive the London equation and magnetic penetration depth. They show that these holographic superconductors are Type II, meaning they transition from a normal phase to a superconducting phase as the magnetic field is reduced. The authors also compute the conductivities of the system, including electric, thermal, and thermoelectric conductivities. They find that the conductivity exhibits a Drude peak at zero frequency, which is a signature of translationally invariant systems. The authors also discuss the connection between superconductivity and conformal field theories, and show that the AdS/CFT correspondence can be used to describe superconductivity in non-conformal field theories. The paper concludes with a summary of the results and some open questions.