In this memoir we present a new construction and new properties of the YangMills measure in two dimensions. This measure was first introduced for the needs of quantum field theory and can be described informally as a probability measure on the space of connections modulo gauge transformations on a principal bundle. We consider the case of a bundle over a compact orientable surface. Our construction is based on the discrete YangMills theory of which we give a full acount. We are able to take its continuum limit and to define a pathwise multiplicative process of random holonomy indexed by the class of piecewise embedded loops. We study in detail the links between this process and a white noise and prove a result of asymptotic independence in the case of a semisimple structure group. We also investigate global Markovian properties of the measure related to the surgery of surfaces. Readership Graduate students and research mathematicians interested in geometry, topology, and analysis. Table of Contents  Discrete YangMills measure
 Continuous YangMills measure
 Abelian gauge theory
 Small scale structure in the semisimple case
 Surgery of the YangMills measure
 Bibliography
