Do You Have to Make up Missed Fasts before Next Ramadan?
Question

I did not fast some days in Ramadan because of my period. This happened a few years ago, and I have not fasted these days until now. What do I have to do?
Praise be to Allah.Obligation to make up missed fasts before next Ramadan 
The imams are agreed that whoever does not fast some days of Ramadan has to make up those days before the next Ramadan comes . 
They quoted as evidence for that the hadith narrated by al-Bukhari (1950) and Muslim (1146) from ‘Aishah (may Allah be pleased with her) who said: “I used to have fasts that I still owed from Ramadan, and I could not make them up until Sha’ban, and that was because of the position of the Messenger of Allah (peace and blessings of Allah be upon him) [as a husband].” 
Al-Hafiz said: 
“It may be understood from her keenness to fast them in Sha’ban that it is not permissible to delay making up fasts until the next Ramadan begins.” 
Scenarios of delaying making up missed fasts until next Ramadan
If a person delays making up fasts until Ramadan begins, one of the following two scenarios must apply.  

The delay is for a reason, such as being sick and the sickness lasting until the following Ramadan begins. There is no sin on a person for delaying in this case, because he has an excuse, and he only has to make up for the days missed. So he should make up the number of days that he did not fast. 
There is no reason for the delay, such as when a person was able to make up the fasts but he did not do so before the following Ramadan began. 

This person is sinning by failing to make up the fasts with no excuse. 
Delaying making up fasts until next Ramadan: Is fidyah a  must?
The imams are agreed that he must make up the fasts, but they differed as to whether along with making up the fasts he must also feed one poor person for each day or not. 
Malik, al-Shaf’i and Ahmad said that he must feed a poor person , and they quoted as evidence for that the reports narrated from some of the Sahabah such as Abu Hurayrah and Ibn ‘Abbas (may Allah be pleased with them). 
Imam Abu Hanifah (may Allah have mercy on him) was of the view that he does not have to feed a poor person as well as making up the fasts. 
He quoted as evidence the fact that Allah commands the one who does not fast in Ramadan only to make up the missed fasts , and He did not mention feeding a poor person . Allah says (interpretation of the meaning): 
“and whoever is ill or on a journey, the same number [of days which one did not observe Sawm (fasts) must be made up] from other days.” [al-Baqarah 2:185] (See al-Majmu’, 6/366; al-Mughni, 4/400)
This second view was also favoured by Imam al-Bukhari (may Allaah have mercy on him). He said in his Sahih: 
Ibrahim (i.e., al-Nakha’i) said: If a person neglects (to make up missed fasts) until the next Ramadan comes, he should fast the missed days of both months. But he did not think that he has to feed a poor person. There was also a mursal report from Abu Hurayrah and Ibn 'Abbas which says that he should also feed a poor person. Then al-Bukhari said: But Allah does not mention feeding a poor person, rather He says (interpretation of the meaning): 
“…the same number [of days which one did not observe Sawm (fasts) must be made up] from other days.” [al-Baqarah 2:185]
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said, when noting that it is not obligatory to feed a poor person: 
“With regard to the words of the Sahabah, their use as evidence is subject to further discussion if it goes against the apparent meaning of the Quran. In this case, saying that it is obligatory to feed a poor person goes against the apparent meaning of the Quran, because Allah only enjoined making up the same number from other days, and He did not mention more than that. Based on this, we should not oblige the slaves of Allah to do any more than they need to fulfil their duty. But what was narrated from Ibn 'Abbas and Abu Hurayrah may be interpreted as referring to what is mustahabb, not what is obligatory. The correct view with regard to this issue is that a person does not have to do anything more than making up the missed fasts, but he is sinning if he delays doing so.” (Al-Sharh al-Mumti’, 6/451) 
Based on this, then what is obligatory is to make up the fasts only, but if a person wants to be on the safe side and feed one poor person for each day missed, then that is good. 
The woman who asked this question – if she delayed making up the fasts with no excuse, has to repent to Allah and resolve not to repeat this mistake in the future. 
And Allah is the One Whom we ask to help us to do that which He loves and which pleases Him. 
And Allah knows best.