  A Hamilton-Jacobi equation with Caputo's time-fractional derivative of order
less than one is considered. The notion of a viscosity solution is introduced
to prove unique existence of a solution to the initial value problem under
periodic boundary conditions. For this purpose comparison principle as well as
Perron's method is established. Stability with respect to the order of
derivative as well as the standard one is studied. Regularity of a solution is
also discussed. Our results in particular apply to a linear transport equation
with time-fractional derivatives with variable coefficients.
