Type Ia supernovae (SNe Ia), widely believed to result from thermonuclear explosions of white dwarfs (WDs), are extremely important phenomena in the universe, owing to their role as distance indicators in cosmology. However, many of their fundamental aspects, e.g., how their progenitors evolve and explode, remain elusive. In particular, it is extensively debated whether the majority of SNe Ia originate from the "single-degenerate (SD) channel" where a WD accretes matter from a non-degenerate companion, or the "double-degenerate (DD) channel" where the explosion is triggered by the dynamical merger of two WDs. In the first half of this talk, I will review recent progress in observational studies toward solving this SD-vs-DD issue. In the latter half, I will focus on X-ray observations of SN Ia remnants, which allow us to measure abundances of heavy elements that were synthesized during the SN explosion. One of the most remarkable results in the last few years from SN Ia remnants would be the detection of strong emission of the Fe-peak elements (Cr, Mn, Fe, Ni) from 3C 397 (Yamaguchi et al. 2015). The measured mass ratios of Mn/Fe and Ni/Fe are the highest reported in any Type Ia SNe or remnants, and require a significant contribution from electron capture that takes place in the dense core of an exploding WD. Recent theoretical work (Dave et al. 2017; Leung & Nomoto 2018) provided an excellent interpretation on this observational result; the progenitor of 3C 397 must have had a central density of 5-6 x 10^9 g/cm^3, much higher than that thought to be standard (~2 x 10^9 g/cm^3). Such high density can be achieved only in the SD system involving a high-mass initial WD (~1M_sun, immediately before the mass accretion starts) and a low-mass companion (~2M_sun). Thus our observation even constrains evolution of the progenitor binary system. I will discuss the implication of this result for the SN Ia progenitor issue and the galactic chemical evolution.