Please read this disclaimer first: The following information has been obtained from sources deemed to be reliable.  However, it is not intended to be legal advice.  We strongly suggest that you verify any of this information with appropriate authority before acting on it.  Where possible, links to official web sites and/or contact information has been supplied to facilitate getting more information.

Please note that firearms and self-defense laws can vary significantly between jurisdictions, the laws can be applied inconsistently from one area to the next, and that those who enforce the laws don't always have as thorough an understanding of the law as one might wish.

Always check with local authority.

See the Arizona Dept. of Public Safety CCW Unit's reciprocity/recognition page for the latest information.

The following states have reciprocity agreements in place with Arizona.  Generally, persons who have an Arizona CCW Permit may legally carry in those states, and persons who hold permits from those states may legally carry in Arizona.

Alaska
Arkansas
Texas
Utah
Kentucky
Michigan.

Some states unilaterally recognize Arizona permits:

Idaho
Indiana Note: Scroll down to IC 35-47-2-21b
Michigan
Montana
Wyoming 
Update 1: Wyoming NO LONGER HONORS ARIZONA PERMITS. 
Update 2: With the new recognition changes to AZ law, WY probably will recognize AZ permits, now that we recognize theirs.  Maybe.  Soon...

Update: As a result of changes to ARS 13:3112 which took effect 18 Sept 2003, AZ now recognizes CCW permits from the following states: Alaska*, Arkansas*, California, Colorado, Connecticut, Delaware, Florida, Iowa, Kentucky*, Louisiana, Maryland,  Massachusetts, Michigan*, Minnesota, Missouri, Montana, Nevada, New Mexico, North Carolina, North Dakota, Oklahoma, Oregon, South Carolina, Tennessee, Texas*, Utah*, West Virginia, Wyoming
*Indicates states with existing reciprocity agreements.

Update: 1 November 2003

The following states have stated they will honor AZ permits: (includes existing reciprocity states)

AK, AR, CO, FL, ID, IN, KY, MI, MT, NC, OK, TN, TX, UT, VA  More states are pending.

Update: 11-21-03, North Dakota has announced it will recognize the Arizona CCW permit
Update: Virginia has canceled recognition of AZ CCW permits.

Additionally, some states allow open carry, and/or allow firearms to be carried in vehicles without any form of permit. As well, some states issue permits to non-residents.  The best source for that information I've been able to find is Packing.org.