  We investigate basic properties of mappings of finite distortion $f:X \to
\mathbb{R}^2$, where $X$ is any metric surface, i.e., metric space homeomorphic
to a planar domain with locally finite $2$-dimensional Hausdorff measure. We
introduce lower gradients, which complement the upper gradients of Heinonen and
Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We
extend the Iwaniec-\v{S}ver\'ak theorem to metric surfaces: a non-constant $f:X
\to \mathbb{R}^2$ with locally square integrable upper gradient and locally
integrable distortion is continuous, open and discrete. We also extend the
Hencl-Koskela theorem by showing that if $f$ is moreover injective then
$f^{-1}$ is a Sobolev map.
