I haven't had a chance to post my shiurim from the previous weeks. I decided I'll start with this week and work backwards.
Rashi quotes the medrash that Sorah had no sins at age 100 just like she had none at age 20. The medrash says that the reason Sorah had no sins at age 20 is because less than 20 a person is not punished b'dinei shamayim. The question is what does this mean and are there other m'koros besides this medrash.
The gemara in Shabbos 89B mentions this idea. Hashem wanted to destroy the B'nei Yisroel and part of Yitzchak's arguement in favor of the B'nei Yisroel was that for the first 20 years of a person's life they don't get punished.
The Rambam in Sanhedrin Perek 7 peirush hamishnayos also says we have a mesorah that under 2o there is no chiyuv kareis.
The Mizrachi in Chayei Sora also writes that this halacha applies to kareis. He says even though the Rambam mentions that one is who over 13 is chayav for malkos and kareis, he means that there is only a chiyuv malkos from age 13.
The Noda B'Yehuda and the Chida in Torah Lishma explain that this halacha applies to all chiyuvim b'dinei shamayim and not just kareis. However, they both point out that it applies b'olam hazeh. A person under 20 will not get punished in this world b'dinei shamayim. Once a person dies, you will get punished for aveiros you did under 20. Furthermore, you will also be held responsible for the aveiros you did as a koton, unless you did teshuva when you got older.
Finally, the CHasam Sofer and Chacham Tzvi both reject this whole idea. The Chasam Sofer says we don't find anywhere in halacha that a person under 20 is not punished. The gemara in Shabbos is aggadata and we don't learn from aggadata.