In this talk I will describe the current renaissance of tests of gravity theories with binary pulsars. I start by describing the classic radiative test of general relativity using the Hulse-Taylor pulsar and the (now) far superior tests done in the double pulsar system. I then introduce the new tests now being done in pulsar-white dwarf systems, which are even more powerful as probes of alternative theories of gravity. Finally, I introduce the new tests with the PSR J0348+0432 binary system. These probe the properties of gravity in a previously unexplored regime of extreme gravitational fields, where new, strongly non-linear effects like spontaneous scalarization (predicted by some alternative theories of gravity, like Scalar-Tensor) can now, for the first time, be constrained. I then list several theories of gravity recently falsified by these experiments, and its consequences for our understanding of dark matter, the fundamental nature of gravitation and the detection of gravitational waves. Finally, I describe the prospects improving gravitational tests with the current set of binary systems and the prospects for testing gravity theories with the recently described millisecond pulsar in a stelllar triple system, PSR J0337+1715.