We propose the Gauge Theory Bootstrap, a method to compute the pion S-matrix that describes the strongly coupled low energy physics of QCD and other similar gauge theories. The method looks for the most general S-matrix that matches at low energy the tree level amplitudes of the non-linear sigma model and at high energy, QCD sum rules and form factors. We compute pion scattering phase shifts for all partial waves with angular momentum $\ell<=3$ up to 2 GeV and calculate the low energy ChiPT coefficients. This is a theoretical/numerical calculation that uses as only data the pion mass $m_\pi$, pion decay constant $f_{\pi}$ and the QCD parameters $N_c=3$, $N_f=2$, $m_q$ and $\alpha_s$. All results are in reasonable agreement with experiment. In particular, we find the $\rho(770)$, $f_2(1270)$ and $\rho(1450)$ resonances and some initial indication of particle production near the resonances. The interplay between the UV gauge theory and low energy pion physics is an example of a general situation where we know the microscopic theory as well as the effective theory of long wavelength fluctuations but we want to solve the strongly coupled dynamics at intermediate energies. The bootstrap builds a bridge between the low and high energy by determining the consistent S-matrix that matches both and provides, in this case, a new direction to understand the strongly coupled physics of gauge theories. Based on work with Martin Kruczenski.