  The Fleming-Viot (FV) process is a measure-valued diffusion that models the
evolution of type frequencies in a countable population which evolves under
resampling (genetic drift), mutation, and selection. In the classic FV model
the fitness (strength) of types is given by a measurable function. In this
paper, we introduce and study the Fleming-Viot process in random environment
(FVRE), when by random environment we mean the fitness of types is a stochastic
process with c\`adl\`ag paths. We identify FVRE as the unique solution to a so
called quenched martingale problem and derive some of its properties via
martingale and duality methods. We develop the duality methods for general
time-inhomogeneous and quenched martingale problems. In fact, some important
aspects of the duality relations only appears for time-inhomogeneous (and
quenched) martingale problems. For example, we see that duals evolve backward
in time with respect to the main Markov process whose evolution is forward in
time. Using a family of function-valued dual processes for FVRE, we prove that,
as the number of individuals $N$ tends to $\infty$, the measure-valued Moran
process $\mu_N^{e_N}$ (with fitness process $e_N$) converges weakly in
Skorokhod topology of c\`adl\`ag functions to the FVRE process $\mu^e$ (with
fitness process $e$), if $e_N \rightarrow e$ a.s. in Skorokhod topology of
c\`adl\`ag functions. We also study the long-time behaviour of FVRE process
$(\mu_t^e)_{t\geq 0}$ joint with its fitness process $e=(e_t)_{t\geq 0}$ and
prove that the joint FV-environment process $(\mu_t^e,e_t)_{t\geq 0}$ is
ergodic under the assumption of weak ergodicity of $e$.
