The first answer is not actually correct. You can claim an exemption for a nonresident spouse, withhout a SSN. And there is no requirement that he live in the US. To file with a nonresident spouse, you must file a W-7 application to obtain an ITIN number for him, this application should be sent in with the 1040 form you file, to the address shown in the w7 instructions.
(There are variabe ways to get apply for this number, see the corresponding instructions for this form)
Your filing status options:
You may have filed HOH in the past, but will not be able to claim his exemption on a HOH return. This is because he does not qualify as a dependent as under IRS rules, a spouse can never be a dependent. And, as well there is no where on a HOH return for claiming a spousal exemption. Your only way to claim him would be to claim his as a "qualifying person", unfortunately nonresident aliens do not qualify as qualifying persons. (These rules are explained in publication 501, p. 7, under the heading "Nonresident Alien spouse" . Available on the IRS website, Internal Revenue Service
You can file using HOH, but will not be able to claim him on that return. In order for a married person to file HOH, they must meet certain tests described in that publication reference. One of which is; you could not have lived in the same household during the last six months of the year. Ill let you look at those rules.
However, you would you maybe able to claim him on a MFS or MFJ return.
To claim him on a MFS return; he must not have any US income, and not be claimed as a dependent on any other US return. He appears to meet this. If you want this filing, again, submit the W-7 application with the 1040/1040A form to get his ITIN number, and mark the return accordingly. The IRS will assign the number during processing the w7 application to the tax return where the SSN goes, and then finish processing the return. Then they'll send a letter with the number, to the address on the application. This is a simple alternative, but one cannot claim EIC and some other credits using this filing status.
For a US citizen, (or resident) to file MFJ with a nonresident, the nonresident must agree to be "treated as a resident" for income tax purposes. This is critical decision, as this means the nonresident has to report worldwide income on this return, even if its earned out of the US. He maybe able to take the foreign earned income exclusion or foreign tax credit if he qualifies , to reduce the impact, or even exclude some or all of it off the return, but it stll must be reported. Additionally a statement must be attached to the return stating he agrees to be treated as a resident for the year, and other genreral information as required. This information, and the statement requirements are stated in Publication 519, p. 10, under the "Nonresident Spouse Treated as a Resident".
I hope this helps answer your question. This rule applies to any US citizen or resident who is married to a nonresident, whether they are in the US or out of the US.
Helpful References: Pub 501 is good for filing status information, but directed towarad US citizens.
Pub 519, Tax Guide for Aliens
Pub 54 Us tax guide for us cittzens and resident aliens abroad.