  Toda Conformal Field Theories (CFTs hereafter) are generalizations of
Liouville CFT where the underlying field is no longer scalar but takes values
in a finite-dimensional vector space. The algebra of symmetry of such models is
given by $W$-algebras, which contain the Virasoro algebra as a subalgebra. In
contrast with Liouville CFT, and in the presence of a boundary, there may exist
non-trivial automorphisms of the associated $W$-algebra, corresponding to
different boundary conditions for the field of the theory.
  Based on this particular feature, we provide in this document a probabilistic
construction of Toda CFTs on a Riemann surface with or without boundary. To be
more specific we define different classes of models in relation with the
different types of boundary conditions suggested by the non-triviality of the
automorphism group of the $W$-algebra. To do so, we rely on a probabilistic
framework based on Gaussian Free Fields and Gaussian Multiplicative Chaos, and
make sense of "Cardy's doubling trick" to construct the underlying field of the
theory.
