  Functionals of Brownian/non-Brownian motions have diverse applications and
attracted a lot of interest of scientists. This paper focuses on deriving the
forward and backward fractional Feynman-Kac equations describing the
distribution of the functionals of the space and time tempered anomalous
diffusion, belonging to the continuous time random walk class. Several examples
of the functionals are explicitly treated, including the occupation time in
half-space, the first passage time, the maximal displacement, the fluctuations
of the occupation fraction, and the fluctuations of the time-averaged position.
