This text defines and studies a class of stochastic processes

indexed by curves drawn on a compact surface and taking

their values in a compact Lie group. We call these processes

two-dimensional Markovian holonomy fields. The proto-type

of these processes, and the only one to have been constructed

before the present work, is the canonical process

under the Yang-Mills measure, first defined by Ambar Sengupta

and later by the author. The Yang-Mills measure sits

in the class of Markovian holonomy fields very much like the

Brownian motion in the class of Lévy processes. We prove

that every regular Markovian holonomy field determines a

Lévy process of a certain class on the Lie group in which

it takes its values, and we construct, for each Lévy process

in this class, a Markovian holonomy field to which it is associated.

When the Lie group is in fact a finite group, we

give an alternative construction of this Markovian holonomy

field as the monodromy of a random ramified principal

bundle. Heuristically, this agrees with the physical origin

of the Yang-Mills measure as the holonomy of a random

connection on a principal bundle.