Given a divisor in a complex manifold, a meromorphic line bundle is a line bundle which becomes singular in a particular way along the divisor. I will describe recent work in which we give a geometric interpretation of such line bundles, classify them using Mixed Hodge structures, and provide new methods of constructing them explicitly. I will also describe the way in which these solve the "prequantization" problem for Logarithmic Picard algebroids. This is ongoing joint work with Kevin Luk.