Models of thermonuclear burning on accreting neutron stars predict different ignition regimes, depending mainly on the mass accretion rate per unit area. For more than three decades, testing these regimes observationally has met with only partial success. I will present recent results from the Fermi-GBM all-sky X-ray burst monitor, which is yielding robust measurements of recurrence time of rare and highly energetic thermonuclear bursts at the lowest mass accretion rates. I will also present RXTE observations of thermonuclear bursts at high mass accretion rates, including the discovery of millihertz quasi-periodic oscillations and several bursting regimes in a neutron star transient and 11 Hz X-ray pulsar. This unusual neutron star, with higher magnetic field and slower rotation than any other known burster, showed copious bursting activity when the mass accretion rate varied between 10% and 50% of the Eddington rate. I will discuss the role of fuel composition and neutron star spin in setting the burst properties of this system, and the possible implications for the rest of thermonuclear bursters.