Type Ia supernovae (SNe) have been used as standardizable candles to discover the accelerated expansion of the Universe; however, we still are not certain how Type Ia SNe exploded exactly. Two popular scenarios have been suggested, one involves a white dwarf (WD) having accreted too much material from a normal-star companion, and the other involves merger of two WDs. In the former scenario, the stellar companion survives the SN explosion, while in the latter scenario no stellar remnant is left. We have been studying the Type Ia supernova remnants (SNRs) in our neighboring galaxy the Large Magellanic Cloud (LMC), using Hubble Space Telescope images to search for surviving companions of their SN progenitors and dense circumstellar material ejected by the SN progenitors. I will report our findings and suggest possible explosion mechanisms. We have also searched for young Type Ia SNRs in the M33 galaxy and found none. This is puzzling because M33 has more stars than the LMC, in which 5 young Type Ia SNRs have been identified. The goal of this research is to understand the explosion mechanisms of Type Ia SNe.