Luke 17:11-19 is the story of a faith that makes possible the healing of a leper and that same faith's recognition of who Jesus is. Here we find a clear instance of faith that is turned towards Jesus. The importance of this topic will become clear towards the end of the Lucan insertion. On the third phase of Jesus' journey to Jerusalem, we will find events, stories and pronouncements that underline the significance of Jesus' person in relation to God and to His kingdom.
The episode begins with a reminder that Jesus is still on his way to Jerusalem (v. 11). We saw the same phrase before in Luke 10:38 as the episode about "the one thing necessary" was introduced. Between 10:38 and Luke 17:10 are stories, events and pronouncements that deal with the things that cause anxiety: hospitality, the fact of discipleship itself, wealth and a secure future, the anxiety of losing someone/something. All of these are somehow balanced with "the one thing necessary" -- Jesus himself. This time, the phrase about the continuing journey towards Jerusalem is followed by the episode involving ten lepers, a story unique to Luke.
Nine Jews (Galileans) and a Samaritan make up the group of lepers. Jews and Samaritans do not normally mix, but the severity of the life of a leper has bonded these men together. When they meet Jesus, they ask for his compassion. They call him by a title that is found in Luke alone. 
Epistata. The title is used in Luke 5:5,8:24.45,9:33.49 and means "Teacher". It is the Greek equivalent for the title Rabbi and a synonym for another title used in the other gospels which means "teacher", 
(didaskale, vocative of 
). Their prayer, "Have mercy on us" is the refrain Catholics use at the beginning of the eucharist, just before the "Gloria".
Jesus tells them to show themselves to the priest. This is the levitical prescription for lepers before they are admitted back into the community (cf.Lev. 14:2-32). Thus, the command of Jesus is not for the healing of the lepers but for their admission into the community. He commends them to show themselves to the priest because the process of healing has already begun when the lepers asked "have mercy on us" and Jesus "saw" them.
The real meat of the story however begins when one of the lepers "sees" that he has been cured. And seeing, he returned. "And he returned with a great cry, praising God." This Lucan phrase echoes the way the shepherds who witnessed the birth of Jesus went home praising and singing hymns to God (Luke 2:2). It is also the phrase that describes the way Naaman the leper went home after being cleansed of his disease at the river Jordan (2 Kings 5:15). The verb "to return" here is the equivalent of the Hebrew 
a 180 degree turn-around that also connotes a conversion. The leper returns to Jesus praising God and therefore implicitly attributing the healing to God himself.
The question of Jesus towards the end of the episode highlights the significance of the man's return.
"Ten were cleansed were they not? Where are the other nine?
Has none but this foreigner returned to give thanks to God?"
Right at the beginning of his ministry, Jesus had already called attention to the "foreigners". In Luke 4:24-27, he mentions the miracles done among foreigners during the time of Elijah and Elisha only to point out that that his townsmates should not expect any miracles from him for their benefit. For this he was nearly lynched (Luke 4:29). Again in Luke 7:9, we find Jesus praising the faith of a Roman centurion while at the same time saying that he has not found an equal faith among the Israelites. The theme of the "foreigner's faith" is one of the sub-themes of Luke-Acts and prepares the way for the narration of the movement of the Gospel towards the Gentiles in Acts.
The words that Jesus tells the man -- "your faith has healed you" -- is the same word pronounced over the sinful woman (Luke 7:50) and the woman cured of hemorrhage (Luke 8:48). Later, he will be saying the same to the blind beggar in Jericho (Luke 18:42). In the first case, it was faith that allowed the woman to be forgiven and therefore be saved. In the case of the woman with hemorrhage and the leper, their faith got their cure, and therefore readmission into the community. In the last case, the blind man is given his place among those who follow Jesus. In all of these cases, we find a "salvation" that is made possible by faith in the person of Jesus.
Right at the beginning of this last stage in the journey towards Jerusalem, a journey that will transmute into a journey towards the ends of the earth after the Resurrection, we find the figure of Jesus at the center of a theme already announced in prophecy of Zechariah: "the experience of salvation" (cf. Luke 1:77). As this last phase of the journey towards Jerusalem reaches its climax on the cross, we will realize that God's salvation is Jesus himself.
For an outline of Luke 17:11-19:27, read this article
