Is my father-in-law a mahram?

Yes. Your husband's father is your mahram. He becomes one as soon as your marriage contract (nikah) is done, and he stays your mahram for life, even after a divorce or your husband's death. That means you can't marry him, you don't need full hijab in front of him, and he can be your mahram for travel.
Why he is a mahram
A mahram is a man you can never marry. Some are mahram by blood (father, brother, son), some by nursing, and some by marriage. Your father-in-law falls in the third group. The Quran names "the wives of your sons" among the women a man may never marry (Quran 4:23), so once you marry his son, you are permanently off-limits to him, and he to you.
The Quran also lists him by name among the men a woman may relax her hijab in front of:
The same applies to your husband's grandfathers on both sides. See Who can see me without my hijab?
Does it end after divorce?
No. All four Sunni schools (Hanafi, Maliki, Shafi'i, Hanbali) agree that this kind of in-law relationship is permanent. The nikah alone creates it; consummation isn't needed. If you divorce or are widowed, your former father-in-law is still your mahram. He just isn't family by name anymore, so use common sense about how close you stay.
What this means day to day
- Hijab: he can see what your father can usually see at home: hair, neck, forearms. Modest clothing still applies.
- Handshakes and hugs: a normal greeting is fine.
- Being alone together: allowed, like with your own father.
- Travel: he can be your mahram on a trip. Some scholars add that a travel mahram should be a trustworthy adult, and a few are cautious when he isn't Muslim.
For more on living with your husband's family, see Do I have to wear hijab in front of my in-laws?
Which in-laws are NOT mahram?
Your husband's brothers, cousins, uncles and nephews are not your mahram. The Prophet ﷺ was asked about a husband's relatives entering upon a wife and said, "The in-laws of the wife are death itself" (Sahih al-Bukhari 5232), a strong warning about the brother-in-law in particular, because people treat him casually. Keep normal hijab and avoid being alone with him. See Can I hug my relatives at holiday gatherings as a Muslim?
For men, it works the same way in reverse: your wife's mother is your mahram for life, while your wife's sisters are not.
- Your husband's father (and grandfathers) are your mahram.
- It starts with the nikah and lasts for life, even after divorce.
- He can see you as your father does and can be your travel mahram.
- Brothers-in-law are not mahram.
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