  The main purpose of this paper is to carry out some of the foundational study
of $C^0$-Hamiltonian geometry and $C^0$-symplectic topology. We introduce the
notions of the strong and the weak {\it Hamiltonian topology} on the space of
Hamiltonian paths, and on the group of Hamiltonian diffeomorphisms. We then
define the {\it group} $Hameo(M,\omega)$ and the space $Hameo^w(M,\omega)$ of
{\it Hamiltonian homeomorphisms} such that $$ Ham(M,\omega) \subsetneq
Hameo(M,\omega) \subset Hameo^w(M,\omega) \subset Sympeo(M,\omega) $$ where
$Sympeo(M,\omega)$ is the group of symplectic homeomorphisms. We prove that
$Hameo(M,\omega)$ is a {\it normal subgroup} of $Sympeo(M,\omega)$ and contains
all the time-one maps of Hamiltonian vector fields of $C^{1,1}$-functions. We
prove that $Hameo(M,\omega)$ is path connected and so contained in the identity
component $Sympeo_0(M,\omega)$ of $Sympeo(M,\omega)$. In the case of an
orientable surface, we prove that the {\it mass flow} of any element from
$Hameo(M,\omega)$ vanishes, which in turn implies that $Hameo(M,\omega)$ is
strictly smaller than the identity component of the group of area preserving
homeomorphisms when $M \neq S^2$. For the case of $S^2$, we conjecture that
$Hameo(S^2,\omega)$ is still a proper subgroup of $Homeo^\omega_0(S^2) =
Sympeo_0(S^2,\omega)$.
