Crouzeix’s Conjecture is an open conjecture which claims that the operator norm of a polynomial applied to a matrix is bounded above by 2 times supremum of the polynomial over the numerical range of the matrix. In this talk I will first give a historical background on Crouzeix’s Conjecture starting from the von Neumann inequality, and then present some new recently published results on the conjecture, which ultimately improve the best known constant for which the conjecture holds on the space of matrices of a fixed dimension.