  Oscillators are at the heart of optical lasers, providing stable transform
limited pulses. In contrast, X-ray free electron lasers use self-amplified
spontaneous emission (SASE), resulting in large stochastic intensity and
spectral fluctuations. Amplified spontaneous emission (ASE) of the $K\alpha_1$
line has been recently observed for Ne gas, Cu compounds and Mn solutions at
the LCLS and SACLA X-ray free electron lasers (XFELs), using an X-ray SASE
pulse as a pump to create population inversion. Here we describe the physics
and realization of an X-ray laser oscillator (XLO) based on periodically
pumping a Cu compound gain medium in a tunable Bragg cavity with a SASE pulse
train, generating intense ($\sim$ 5 x 10$^{10}$ ph/pulse), fully coherent,
transform limited 8 keV pulses with 48 meV spectral resolution. We also discuss
extending these results to other elements to operate XLO from about 5 to 12
keV, improving X-ray-based research beyond current capabilities.
