Separation & Isolation
The Tenenbaum family does not exist in a whole, stable state for long. Early on in the movie, we see that Royal's careless ways lead to his separation from his wife Etheline, which creates the emotional and circumstantial backdrop for the whole film. In leaving his wife, Royal is also leaving his children, separating from them in a rather unceremonious way. He sits his precocious children around their dining room table and straightforwardly tells them that he has to leave. Not only does Royal physically separate himself from his family, but he is also very psychically and emotionally separated from them as well. This attitude creates a familial dynamic in which they all separate from one another, and never experience any real closeness.
Child Prodigies & Failure
The Tenenbaum children are preternaturally gifted in different areas: Richie in athletics, Margot in writing, and Chas in business. This makes them very special children, and as the narrator tells us, Etheline valued their education and the cultivation of their talents above all else. However, the film shows that these early childhood gifts do not age well. As adults, all three of the Tenenbaums are incredibly neurotic failures, each living passively in the wake of their childhood gifts. Thus, a major theme of the film is the shadow that early success casts over the life of a child prodigy. The fact that the Tenenbaums are robbed of a childhood makes them broken adults, and the film depicts their arrested development and inability to deal with failure in tragicomic ways.
Fatherhood
Royal Tenenbaum is the patriarch of the Tenenbaum family, and in many ways, the protagonist of the film. He is a lovable jerk who doesn't quite know how to be part of his family, and in his dysfunctional approach to love and affection, he stands in for a broader representation of fatherhood. While he knows how to show his sons and grandsons how to live on the edge and experience life to the fullest (going to dogfights, riding on the back of garbage trucks, indulging in hamburgers almost daily), he doesn't really know how to show up for them emotionally, and he treats Margot, his adopted daughter, almost like a stranger. Royal's failed attempts at being a father are often jokes in the film—indeed, his oversights and failures as a father are at times so obvious as to seem absurd—but they also represent something about the difficulties of fatherhood, the emotional detachment that a father can have from a child, and the difficulties that a father (especially a bad one) has in connecting and finding emotional rapport with his family.
Love
Even though he has been separated from his wife for 22 years, Royal becomes incredibly jealous when he hears that Etheline is getting re-married. Even if he doesn't feel particularly attached to his ex-wife, he wants to prove that she belongs to him. In contrast, Henry Sherman, Etheline's suitor and fiancé, is patient and loving towards her, and the affection and care shared between the new lovebirds stands in stark contrast to the deceptive posturing of Royal Tenenbaum. In this way, love is a theme in the film, as reflected in the affection shared between Henry and Etheline.
More complicatedly, Richie and Margot are in love. This love is taboo, because they are legally siblings. Margot was adopted, so there is nothing biologically making their romantic attachment incestuous, but their having grown up as siblings makes their connection less acceptable. As hard as they try to fight it, however, the couple shares a pure and genuine attachment to one another. Their love is too strong for them to simply ignore it, and they eventually give in to it by the end, with the blessing of the people around them. In spite of the dark complications swirling around many of the attachments in the film, love is a major theme.
Depression
All of the children in The Royal Tenenbaums are depressed. Chas is anxious and in mourning about his wife's death, Richie suffered a horrible tennis failure and has secluded himself in the middle of the ocean, and Margot does nothing but smoke cigarettes and watch television. The fallout from their promising youths becomes adult depression in the Tenenbaum children. A melancholy shade hangs over nearly all their interactions, a kind of limp deadpan delivery that seems like depression.
One of the climactic moments of the film is when Richie learns that Margot has been having a number of affairs. He goes in the bathroom, cuts off all his hair and then attempts suicide by slitting his wrists. It is a shocking and gruesome display of self harm, and the event seems to take all of the instances of depression in the film and give them a moment of catharsis. He is rushed to the hospital and survives, and the act seems to create some kind of unity and connection around the isolating feelings of sadness felt by all of the Tenenbaums.
Homecoming
Early on in the film, Chas becomes so paranoid about the safety of his own home that he decides to move himself and his two sons into his childhood home. Jealous that Chas is able to do so, Margot comes home as well. In a strange regressive chain, the two Tenenbaum children return to their childhood rooms, perfectly preserved as though they have never left.
In order to escape homelessness, Royal Tenenbaum devises a lie that he is dying of stomach cancer. By pretending to be terminally ill, Royal manages to gather his family around him and snag a room in his old home. At the news of his father's "illness," Richie returns home as well, and for the first time in 22 years, all of the Tenenbaums are under one roof. Thus a major theme of the film is homecoming, not just in the physical sense, but also a return to childhood, to the past, to the preserved family unit.
Redemption
Ultimately, the film is a redemption narrative, in that by the end, Royal has made amends with his family, even if he hasn't changed very much. He shows up for Richie when Richie asks his advice about Margot, he apologizes to Henry for giving him such a hard time, and he reconciles with Chas. While he hasn't necessarily become a markedly better person, he has redeemed himself in the eyes of his family members and figured out how to be more accountable to them.
More broadly, it is a redemption narrative for all the characters. By the end, Chas becomes less fearful, Margot has a play produced, and Richie begins to teach tennis. All of the Tenenbaums learn to integrate themselves more healthily with the world around them, to repent for past mistakes and neuroses, and to find some level of peace.
