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 induced by a complex simple Lie algebra. The goal of this document is to provide a probabilistic construction of such models on all compact hyperbolic Riemann surfaces with or without boundary. To do so, we rely on a probabilistic framework based on Gaussian Free Fields and Gaussian Multiplicative Chaos. In the presence of a boundary, one major difference with Liouville CFT is the existence of non-trivial outer automorphisms of the underlying Lie algebra. The main novelty of our construction is to associate to such an outer automorphism a type of boundary conditions for the field of the theory, leading to the definition of two different classes of models, with either Neumann or Cardy boundary conditions. This in turn has implications on the algebra of symmetry of the models, which are given by $W$-algebras.