In the news last week, multiple conservative Catholic websites (but no liberal ones) have reported on a rumor that Pope Francis intends to hold a synod, on the topic of married priests. The idea is that Pope Francis will announce it when he visits Chiapas, Mexico this February.
Chiapas is a place with a few dozen priests, and hundreds of married deacons. Church authorities thought it was weird that the Bishop there was basically only ordaining deacons for forty years, and so they suspended all deacon ordinations from 2000-2014.
The issue of married priests is an interesting one. Unmarried priests have been around since the time of Jesus, and in many places, it was the standard practice for Bishops to never ordain married men to the priesthood. The general standard of not ordaining married priests became a universal standard, during the reforms of Pope Saint Gregory VII (around 1050 AD).
Obviously, the issue has come up in the past 1,000 years, and there have been times when married men got ordained. The church's general response in these exceptional cases has been mandatory continence. Continence is basically celibacy for married people, the idea being that if a married guy wants to be a priest, he has to be like all other priests and abstain from sex. They use the different word "continence" for this situation, because it's lawful for a married man to have sex with his wife, whereas it's unlawful for a celibate priest to have sex with anyone.
Can the Catholic Church go back to the practice of ordaining married men to the priesthood? Yes. But the next question is whether or not there will be mandatory continence for these priests. People assume the answer is "no", because married deacons aren't required to be continent. Tradition, on the other hand, says "yes". Also, Canon Law 277 says, "yes".
I assume the second question will be, "Can current priests get married?". That answer is no. For one thing, all current priests have sworn an unbreakable promise of celibacy. For another, the order of events has been set in stone, so to speak, by Saint Paul in the Bible (Titus 1:5-9, 1 Timothy 3). You have to be married before becoming a priest. Someone who is already a priest can't be married, and if you've got a widower priest (or deacon), they can't remarry.