The predominant scientific theory regarding the nature of this process involves self-organization, simple self-replicating molecules becoming gradually more complex, and processes much like those seen in evolution.
Although it is consistent with the theory of evolution, neither abiogenesis in general (nor Terran abiogenesis in particular) is part of that theory. Despite this, lack of rigorous evidence for abiogenesis is often used as an argument against evolution and in favor of vague mythological origin stories with no supporting evidence whatsoever.
While most mythologies include a story explaining the origins of (non-supernatural) life on Earth, none have so far proposed a general theory of abiogenesis. Presumably whatever mythological beings or processes first gave rise to life on Earth could also give rise to life on other planets, but this is rarely discussed in the scripture of traditional religions -- presumably because the idea of planets as large solid bodies with atmospheres was not known of when the vast majority of scripture was first written.
Any theory which explains the origin of life by hypothesizing supernatural beings pockets the question, of course, of the origin of those beings – which would be a far more difficult question.