Septuagesima (in full, Septuagesima Sunday) is the name for the ninth Sunday before Easter, the third before Ash Wednesday. The term is sometimes applied also to the period that begins on this day and ends on Shrove Tuesday, the day before Ash Wednesday, when Lent begins. This period is also known as the pre-Lenten season or Shrovetide.
The other two Sundays in this period of the liturgical year are called Sexagesima and Quinquagesima, the latter sometimes also called Shrove Sunday. The earliest date on which Septuagesima Sunday can occur is January 18 (Easter falling on March 22 in a non-leap year) and the latest is February 22 (Easter falling on April 25 in a leap year).
Origins of the term
Septuagesima comes from the Latin word for "seventieth" with Sexagesima and Quinquagesima equalling "sixtieth" and "fiftieth" respectively. They are patterned after the Latin word for the season of Lent, Quadragesima, which means "fortieth", as Lent, not counting Sundays, is forty days long.
Because every Sunday recalls the resurrection of Christ, they are considered "little Easters" and not treated as days of penance. Quinquagesima Sunday is indeed the fiftieth day before Easter (counting inclusively), but the numbers indicated by the names "Sexagesima" and "Septuagesima" do not correspond to the interval between these Sundays and Easter.
Amalarius of Metz would have the name indicate a period of seventy days made up of the nine weeks to Easter plus Easter Week, which would mystically represent the seventy-year Babylonian captivity.
Septuagesima was also the day on which one could begin a forty-day Lenten fast that excluded from its observance Thursdays, Saturdays, and Sundays.
Devotional and liturgical practices
The 17-day period beginning on Septuagesima Sunday was intended to be observed as a preparation for the season of Lent, which is itself a period of spiritual preparation (for Easter). In many countries, however, Septuagesima Sunday still marks the traditional start of the carnival season, culminating on Shrove Tuesday, sometimes known as Mardi Gras.
In the pre-1970 Roman Catholic liturgy, the Alleluia ceases to be said during the liturgy. At first Vespers of Septuagesima Sunday, two alleluias are added to the closing verse of Benedicamus Domino and its response, Deo gratias, as during the Easter Octave, and, starting at Compline, it is no longer used until Easter. Likewise, violet vestments are worn, except on feasts, from Septuagesima Sunday until Holy Thursday. As during Advent and Lent, the Gloria and Te Deum are no longer said on Sundays.
The readings at Matins for this week are the first few chapters of Genesis, telling of the creation of the world, of Adam and Eve, the fall of man and resulting expulsion from the Garden of Eden, and the story of Cain and Abel. In the following weeks before and during Lent, the readings continue to Noah, Abraham, Isaac, Jacob, and Moses. The Gospel reading for Septuagesima week is the parable of the Workers in the Vineyard (Matthew 20:1-16).
The new liturgical books created after the Second Vatican Council omit Septuagesima, Sexagesima and Quinquagesima Sundays. In these, the period encompassing them became part of Ordinary Time, and because of this, the use of violet vestments and omission of "Alleluia" in the liturgy do not start until Ash Wednesday. The traditional liturgical books, such as the Missal of John XXIII and the Roman Breviary, however, continue to include the Septuagesima season.
Polish National Catholic Church usage
The Polish National Catholic Church has officially reinstated the Septuagesima, Sexagesima, and Quinquagesima Sundays in 2014 throughout the entire Church. The celebration of this season as a preparation for Holy Lent is now highlighted as a part of the Liturgical Year. 
Most provinces of the Anglican Communion adopted the same change. In the Church of England these Sundays retain their original designations where the Prayer Book Calendar is followed, but in the Common Worship Calendar they have been subsumed into a pre-Lent season of variable length, with anything from zero to five "Sundays before Lent" depending on the date of Easter. Churches in the Continuing Anglican movement that use the 1928 Book of Common Prayer (or the various missals based upon it) also observe Septuagesima.
Some Lutherans still celebrate this season, however, those who adopted the three-year lectionary following its appearance in the Roman Church, also followed the practice of eliminating Pre Lent as a season and instead continue with the Sundays after the Epiphany, the last of which is celebrated as the Feast of the Transfiguration.
A pre-Lent season also exists in the Eastern Orthodox and Byzantine Catholic liturgical calendar, and is found in the liturgical book known as the Triódion (which continues to Easter Even). It is 22 days long because it begins on the Sunday before Septuagesima, but not 24 since the Byzantine Lent commences on a Monday instead of a Wednesday.