The phrase means undertaking a task despite all seemingly insurmountable opposition or probability. Often the chances of success are very low. Against all odds表示要冲破一切险阻去做某事，尽管这件事情看起来很有可能会失败。

Examples:
Against all odds, she conceived her first child at the age of 35. 冲破一切困难，她在她35岁的时候怀上了她的第一个孩子。
He struggled against the odds to keep his business going during the recession. 在萧条时期，他尽一切努力保全他的公司。