In this dissertation we investigate Lagrange interpolation. Our first result will deal with a hierarchy of interpolation schemes. Specifically, we will show that given a triangular array of points in a regular compact set K, such that the corresponding Lebesgue constants are subexponential, one always has the uniform convergence of Ln(f) to f for all functions analytic on K. We will then show that uniform convergence of Ln(f) to f for all analytic functions f is equivalent to the fact that the probability measures γn = 1/n Σn j=1 δzn,j , which are associated with our triangular array, converge weak star to the equilibrium distribution for K. Motivated by our hierarchy, we will then come to our main result, namely that the Lebesgue constants associated with Leja sequences on fairly general compact sets are subexponential. More generally, considering Newton interpolation on a sequence of points, we will show that the weak star convergence of their corresponding probability measures to the equilibrium distribution, together with a certain distancing rule, implies that their corresponding Lebesgue constants are sub-exponential.