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A homily is a commentary that follows a reading of scripture. In Catholic, Anglican, Lutheran, and Eastern Orthodox Churches, a homily is usually given during Mass (Divine Liturgy for Orthodox and Eastern Catholic Churches, and Divine Service for the Lutheran Church) at the end of the Liturgy of the Word. Many people consider it synonymous with a sermon.
Roman Catholic Mass homily 
The General Instruction of the Roman Missal (GIRM)
29. When the Sacred Scriptures are read in the Church, God himself speaks to his people, and Christ, present in his own word, proclaims the Gospel. Therefore, all must listen with reverence to the readings from God's word, for they make up an element of greatest importance in the Liturgy. Although in the readings from Sacred Scripture God's word is addressed to all people of every era and is understandable to them, nevertheless, a fuller understanding and a greater effectiveness of the word is fostered by a living commentary on the word, that is, the Homily, as part of the liturgical action.
65. The Homily is part of the Liturgy and is strongly recommended, for it is necessary for the nurturing of the Christian life. It should be an exposition of some aspect of the readings from Sacred Scripture or of another text from the Ordinary or from the Proper of the Mass of the day and should take into account both the mystery being celebrated and the particular needs of the listeners.
66. The Homily should ordinarily be given by the priest celebrant himself. He may entrust it to a concelebrating priest or occasionally, according to circumstances, to the deacon, but never to a lay person. In particular cases and for a just cause, the Homily may even be given by a Bishop or a priest who is present at the celebration but cannot concelebrate. There is to be a Homily on Sundays and holy days of obligation at all Masses that are celebrated with the participation of a congregation; it may not be omitted without a serious reason. It is recommended on other days, especially on the weekdays of Advent, Lent and the Easter Season, as well as on other festive days and occasions when the people come to church in greater numbers. After the Homily a brief period of silence is appropriately observed.
The word homily is derived from the Greek word homilia (from homilein), which means to have communion or hold verbal intercourse with a person. In this sense homilia is used in 1 Corinthians 15:33. In Luke 24:14, we find the word homiloun, and in Acts 24:26, homilei, both used in the sense of "speaking with". Origen was the first to distinguish between logos (sermo) and homilia (tractatus). Since Origen's time homily has meant, and still means, a commentary, without formal introduction, division, or conclusion, on some part of Sacred Scripture, the aim being to explain the literal, and evolve the spiritual, meaning of the Sacred Text. The latter, as a rule, is the more important; but if, as in the case of Origen, more attention be paid to the former, the homily will be called expository rather than moral or hortatory. It is the oldest form of Christian preaching.
Other senses 
In certain special occasions in the Episcopal Church including, but not limited to, the traditional Easter Vigil service, homily may be defined as a summoning of young and/or infantile children, as an immediate precursor to a linguistically casual and semi-interactive monologue presented by a bishop or equally highly ranking member of aforementioned clergymen.
In colloquial usage, homily often means a sermon concerning a practical matter, a moralizing lecture or admonition, or an inspirational saying or platitude.
See also 
- Read Malayalam and English Homilies, Reflections and Talks By Archbishop Soosa Pakiam, Metropolitan Archdiocese of Trivandrum
- 2002 General Instruction of the Roman Missal – England and Wales edition (pdf)
- Homily Points
- The Homilies of Father Robert S. Smith – Cornell Catholic Community, Smith's Homilies