Thermonuclear bursts have been observed from accreting neutron stars for 35 years and provide an ideal space laboratory for nuclear physics, high energy astrophysics, and general relativity. Millisecond oscillations observed in the past 15 years with the Rossi X-ray Timing Explorer have also revealed the spin of these objects and the apparent asymmetry in the burning conditions. The latter led to the phenomenological picture of single-point ignition of unstable burning and subsequent spreading over the neutron star surface. This picture, however, cannot explain the observation of oscillations in burst tails and the varying burning coverage of neutron star surface from burst to burst, neither could it yet be established computationally. In the proposed project I aim to resolve some of the key open questions, theoretical and observational, by combining the knowledge that I have gained in my research in the USA with the expertise from members of the Institute of Astronomy and Astrophysics at the University of Tuebingen. In specific, my proposed research plan consists of i) performing timing analysis of the X-ray oscillations observed at the rise of bursts and comparing those with computed waveforms to determine the geometrical properties of the hot-spots, ii) fitting the X-ray spectra observed with RXTE to atmospheric models to determine the chemical composition of the burning material and the evolution of the apparent burning area during bursts, iii) performing 2-dimensional computations of accreted layers for determining the criteria for localized unstable burning, iv) exploring surface modes as an alternative explanation for burst oscillations, and v) studying the X-ray timing noise of accretion- powered millisecond pulsars to study the movements of the accretion column in relation to the burning hot-spot from those sources. These would then place constraints on the nuclear conditions leading to these bursts and the equation of state of neutron stars.