At the heart of many problems in Commutative Algebra and Algebraic Geometry is the difference between symbolic and regular powers of a homogeneous ideal. One way to find failures of containments between these powers is to use an asymptotic approach and look at a special limit called the Waldschmidt constant. This limit was first introduced as a way to estimate the lowest degree of a hypersurface vanishing at all the points of a variety to a given order. However, this useful limit is challenging to compute. We will give some interpretations of the Waldschmidt constant of a monomial ideal which allow us to determine this limit in a number of cases. This is joint work from two projects: the first with R. Embree, H. T. Ha, and A. Hoefel and the second with C. Bocci, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Seceleanu, A. Van Tuyl, and T. Vu.