The classical theory of electrodynamics can indeed be written as a geometrical theory in a similar way to general relativity. As it happens there is a question and answer addressing just this, but it's in the Maths SE: Electrodynamics in general spacetime.
Classical electrodynamics is an example of a class of theories called classical Yang-Mills gauge theories, though Maxwell didn't realise this as the Yang-Mills theories were first described in 1954. These are geometric theories like general relativity, though note that GR is not a Yang-Mills theory - if it was we'd probably have quantised it by now. There are various introductions to Yang-Mills theory around, and a quick Google found this introduction (350KB PDF) that seems pretty good. The theories use a curvature tensor, though this is different to the Riemann tensor used in GR.
Quantising the Yang-Mills classical theory gives quantum electrodynamics i.e. the quantum field theory describing electrodynamics. The weak and strong forces are also quantum Yang-Mills theories, though in these two cases there is no useful classical theory.
Christoph points out in a comment that there is an alternative route to a geometric theory of electrodynamics. In 1919 Theodor Kaluza pointed out that if general relativity was formulated in 5 dimensions (4 space and 1 time) the theory incorporated electrodynamics. This approach was built upon by Oskar Klein and is now known as Kaluza-Klein theory. However the theory requires there to be extra dimensions of space, and in any case the electrodynamic bit of the theory is really a Yang-Mills theory in disguise.