I think they do but have not been able to find out for sure
When did the practice start, or has it always been done? I have read the early Christians said mass right over the graves of the saints, was there a reason for this in scripture?
I know a bit about relics, but I think some of you know a great deal about many things…Can you enlighten me?
Yes there is a relic in each altar. At one time it was a crime punishable by death to be at mass. This is why mass was offered inside the catacombs on the tombs of people who were martyrs. The mass was being offered in secrecy and hidden so not to suffer persecution.
I thought some reform of Vatican II or afterwards allowed for altars without relics. because i asked a parish preist to tell me about the relic in our church’s altar, and he told me he had no idea.
No, there was no such reform. however, it is likely that your priest actually doesn’t know much about the relic, since as I understand, alot of altar relics are anonymous.
Byzantine Holy Tables lack relics. But, as with the Roman church, relics are required to say the Divine Liturgy.
The Byzantine RIte, in its 8 Catholic churches and 20-some-odd orthodox churches, uses an antimension which holds the relics.
The antimension is made of fine fabrics, and is decorated with iconographic representations of the Cross and the Gospel Writers. It has a packet of relics sewn to it, and must be signed by a bishop. In form, it is similar to the Roman corporal: a large square of fabric.
Literally, the functional definition of a byzantine parish is having its own antimension.
Wherever the Antimension is, there is also the bishop’s permission to say the divine liturgy.
One of the major steps for a mission is getting their own antimension. Failing that, a priest from the parent parish may bring the antimension for each liturgy.
I’m not entirely sure all Catholic altars do have relics nowadays, especially the ones that are constructed of wood and made to look similar to normal tables. However, Vatican II never said anything about doing away with the tradition.
Here are two sources of information about altars and relics: GIRM 298-302 and CIC 1235-1237. I’ve snipped out information that is irrelevant to the matter at hand:
GIRM 298. It is appropriate to have a fixed altar in every church … An altar is called **“fixed” if it is attached to the floor **so as not to be removeable; otherwise it is called "moveable."
GIRM 300. An altar whether fixed or movable is dedicated according to the rite prescribed in the Roman Pontifical; but it is permissible for a movable altar simply to be blessed.
GIRM 302. The practice of placing relics of Saints, even those not Martyrs, under the altar to be dedicated is fittingly retained.
Can. 1235 §1 The altar or table on which the eucharistic Sacrifice is celebrated is termed fixed if it is so constructed that it is attached to the floor and therefore cannot be moved; it is termed movable, if it can be removed.
§2 It is proper that in every church there should be a fixed altar. …
Can. 1237 §1 Fixed altars are to be dedicated, movable ones either dedicated or blessed, according to the rites prescribed in the liturgical books.
§2 The ancient tradition of placing relics of Martyrs or of other Saints within a fixed altar is to be retained, in accordance with the rites prescribed in the liturgical books.
So, long story short: it is “proper” and “appropriate” for every church to have a fixed altar, and fixed altars “are to be dedicated” (while moveable altars can simply be blessed), and the tradition of placing relics in the altar to be dedicated “is to be retained”.
My parish doesn’t have a fixed altar (as far as I know), though, so “proper” might be one of those loosey-goosey words. 
I would hope, though, that every fixed altar really does contain relics, since the tradition of doing so “is to be retained”.
The one in our new parish does not have one. Not sure where the one from the old building went and the new one has a fixed altar.
The priest at Saint Mary’s in McKeesport, PA was kind enough to give a lesson to our homeschool group on the relic in the antimension. I believe he said it was given to each priest when he is ordained and that it goes with him.
Did you see the Pope consecrating the new altar, and they put relics in to it before he did?
I thought it was a privilege to be able to see this
Wonderful a once in our life time IMHO. The Cathedral in Sydney would be one reason I would like to visit Australia. The wildlife (animals) is another.
We have a movable Altar, and even it has relics embedded inside. You can tell by the little square stone with cloth on it; pay attention on Good Friday! 
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The most beautiful mass I ever attended was in that Cathedral with Cardinal Pell presiding.
Pax,
OA