The 2-spin spherical Sherrington Kirkpatrick (SSK) model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual Sherrington-Kirkpatrick model with Ising spins. The SSK model admits tractable formulas for many of its observables, allowing for a detailed analysis of its fluctuations using techniques from random matrix theory. We discuss recent results on the fluctuations of the SSK model with a magnetic field as well as at critical temperature. Based on joint work with P. Sosoe.