The Innocence Project and similar efforts have successfully overturned

The Innocence Project and similar efforts have successfully overturned numerous convictions based on DNA evidence that had not been either discovered or presented during trial that tends to negate the defendant's guilt.  In some cases, individuals who were clearly not guilty of the charged offenses have spent many years in prison.  What should be the standard on appeal for inmates (or those who have already completed their incarceration) to gain the opportunity to re-investigate their cases, present new evidence, or otherwise challenge their convictions?  What issues should they be allowed to litigate (DNA, false confession, ineffective assistance of counsel, etc.)?