The narrator begins by saying that the story that will be told in the novel is that of Jean-Baptiste Grenouille, one of the most "gifted and abominable personages" of 18th-century France. The narrator continues that his name has been forgotten (unlike figures such as Napoleon Bonaparte) not because he was any less awful than any of the other abominations of the era, but because he dealt in scent, which doesn't leave a trace.
The idea that scent leaves no trace will be an important idea throughout the novel, and given later events, the reader will be asked to question the actual truth of this. Grenouille is introduced from the start as an abominable figure, which drives the reader away from identifying with him to begin with.
In 18th-century France, everything stank disgustingly in a way the narrator deems inconceivable to the modern person. Rich or poor, summer or winter, every person and every place, particularly Paris, smelled horrible. However, the Cimetière des Innocents, which housed 800 years' worth of bodies from the surrounding parishes and hospitals, stank the worst.
The cemetery mentioned here will be an important location in the novel at several points. Further, notice the narration style here as the narrator describes the stench of the city. The description has a visceral quality to it, despite the assertion that "modern people" couldn't conceive of such a stench.
Jean-Baptiste Grenouille is born here on July 17, 1738. Grenouille’s mother is a fishmonger, and Grenouille is her fifth baby. All her babies were born in her market stall, although none of the others lived. The narrator notes that Grenouille's mother is still young, pretty, and relatively healthy, and entertains a hope of one day marrying a widower and bearing "real children." When she finally gives birth to Grenouille, she cuts the umbilical cord with her butcher knife, and, affected by the heat and the smell of the market and cemetery, faints.
Note here Grenouille's mother's desire to go on and have "real children." Throughout the novel, children are described as sub-human in different ways, and Grenouille in particular is seen as not truly human. This quote then indicates that Grenouille, along with his mother’s previous dead children (who are presumably born out of wedlock, and therefore not seen as “real” or legitimate either), aren't important or worth bearing in the first place. This idea will follow Grenouille throughout his life, as he remains unloved and unvalued.
This draws a crowd and someone calls the police. When Grenouille's mother comes to, she stands up and goes to wash. Grenouille begins to cry under the table in the market stall and the crowd discovers him. He is given to a wet nurse and his mother is arrested, tried for multiple infanticides, and decapitated a few weeks later.
Here we see the judicial system of the time at work. There are very real consequences for (perceived) infanticide, and in a way Grenouille is punished for his mother's wrongdoing by being deprived of any family from the start.
By the time his Grenouille’s mother was decapitated, Grenouille had had three different wet nurses, all of whom said that he was too greedy for milk. La Fosse, the police officer in charge of Grenouille's case, wants to send Grenouille to a halfway house that sends baptized orphans to Rouen, but since Grenouille is at this point still nameless and not baptized, La Fosse instead turns him over to the cloister of Saint-Merri. They baptize him with the name Jean-Baptiste and give him to Jeanne Bussie, a wet nurse, who receives three francs per week to feed him.
Notice the irony here: while Jean-Baptiste was a popular name at the time, the historical John the Baptist was beheaded, just like Grenouille's mother. Grenouille then essentially has to carry his mother's act of wrongdoing with him for his entire life. While this turn is tragic, it's also humorous in a very dark way, setting the novel up for further instances of dark comedy.