  In this paper we study the Dirichlet problem of translating mean curvature
equations over domains in Riemannian manifolds with dimension $n$. Imitating
the generalized solution theory of Miranda-Giusti, we define a new conformal
area functional and a generalized solution to this Dirichlet problem. The
existence of generalized solutions to this problem on bounded Lipschitz domains
is established. If the domain is mean convex and bounded with $C^2$ boundary,
its closure does not contain any closed minimal hypersurface except a singular
set with its Hausdorff dimension at most $n-7$ and the boundary data is
continuous, the generalized solution is the desirable classical smooth
solution. The non-minimal condition could not be removed in general.
