Is it possible to receive the Eucharist under both kinds?

Holy Communion may be received under both kinds, though in receiving either the consecrated host or the consecrated wine, one receives Jesus' body and blood.

In the Latin Rite of the Catholic Church, Holy Communion is most frequently distributed to the people under the form of unleavened bread.

In many Masses, especially during daily Mass, only the priest and deacon receive the Eucharist under the form of consecrated wine.

This does not mean the people are forbidden from receiving both the body and blood of Jesus. Generally speaking it is only a matter of pastoral and practical circumstances.

Precious Blood

The Catechism of the Catholic Church explains that in the Latin Rite, Holy Communion is most commonly distributed under one form:

Since Christ is sacramentally present under each of the species, communion under the species of bread alone makes it possible to receive all the fruit of Eucharistic grace. For pastoral reasons this manner of receiving communion has been legitimately established as the most common form in the Latin rite.
CCC 1390

However, the Church sees the distribution of Holy Communion under both forms as a more complete sign:

But “the sign of communion is more complete when given under both kinds, since in that form the sign of the Eucharistic meal appears more clearly.” This is the usual form of receiving communion in the Eastern rites.
CCC 1390

For various pastoral reasons, bishops in the Latin Rite have preferred to only distribute the consecrated hosts, reserving the Precious Blood for special occasions. Often both forms are distributed at First Communion, Confirmation, and Holy Matrimony.

Yet, many parishes distribute the Precious Blood on a weekly and even daily basis.

One of the reasons some parishes do not distribute the Precious Blood on a regular basis is due to the possibility of spilling the consecrated wine in the handling of the chalice.

Another practical reason is that a parish may not have enough Extraordinary Ministers of the Eucharist present at a Mass, and so it makes it difficult to distribute under both forms.

The key to remember is that Jesus is fully present under either form and should be received with great reverence and love.