How Can I Clear My Arrest Or Conviction Record?

There are several possible ways of accomplishing this. In most cases your record of arrest will remain but the case can show on your live scan as dismissed.
If you have been convicted of a misdemeanor and you have completed probation and have been free of any new offenses for one year you can apply for a dismissal under 1203.4 of the Penal Code. You must obtain a current live scan and submit it with a proper petition and 15-day notice to the District Attorney. Once the petition is granted and your case dismissed you can on a job application instead of reporting a conviction you can respond by saying that you were charged with an offense but the case was dismissed.
If you have been convicted of a felony and granted probation and completed it you are also eligible for the dismissal under 1203.4 of the Penal Code.
If you have been convicted of an infraction or misdemeanor and probation has not been granted you can still get a dismissal as above if one year has passed and you have been free of police contact.
Felony convictions without probation but with county jail sentences are eligible for this relief. However if you served State Prison time you are not eligible for this relief.
If you are actually factually innocent of any crime with a petition filed under 851.8 of the Penal Code you are eligible to have your arrest and conviction record sealed. These petitions are usually denied by police agencies and will require representation by an attorney to set the case for a court hearing.
In the instances where you have been convicted of a state offense and given state prison time with no probation there is a procedure to obtain a Governors Pardon. This requires noticing the District Attorney in the county where you reside and the District Attorney where the offense occurred. A court hearing is scheduled and if the petition is granted it is forwarded to the Governor who may or may not pardon you.
If convicted of a federal offense there is a procedure to obtain a Presidential pardon where special government attorneys are assigned to process these petitions for Presidential pardon.
All of these options require the services of an experienced attorney because of the complex nature of the forms and the requirements of legal notice.