  A classification for Brownian motions on metric graphs, that is, right
continuous strong Markov processes which behave like a one-dimensional Brownian
motion on the edges and feature effects like Walsh skewness, stickiness and
jumps at the vertices, is obtained. The Feller property of these processes is
proved, and the boundary conditions of their generators are identified as
non-local Feller-Wentzell boundary conditions. By using a technique of
successive revivals, a complete description of the generator is achieved for
Brownian motions on star graphs.
