Boardwalk Empire/WMG

Jimmy is the great-grandson of Amsterdam and Jenny from Gangs of New York.
Michael Pitt and Leonardo Dicaprio look similar, Jimmy is Irish-American and Amsterdam & Jenny were Irish immigrants, Jimmy uses the same "knife in the boot" trick that Amsterdam tries, both are young and delicate-looking but very tough, and Atlantic City isn't far from New York. Gangs of New York and Boardwalk Empire are both Martin Scorsese productions with similar tones and themes, and both feature fictional and fictionalized criminal characters rubbing shoulders with real-life historical figures.

Boardwalk Empire takes place in the same 'verse as the Sopranos.
Both focus on the Jersey Mafia.

Boardwalk Empire takes place in the same 'verse as The Wire.
Omar is of course Chalky's Identical Grandson.

Nucky is Jimmy's father, but Jimmy doesn't know it.
Jimmy claims to have no father. We know from the preview for "Broadway Limited" that Nucky has some sort of connection to Jimmy's mother that has led him to watch out for Jimmy.
 * This is officially disproven. See below.

Alternatively, the Commodore is Jimmy's father
Nucky wastes some precious time for no apparent reason to tell his intentions to the Commodore before tossing Jimmy out... despite being quite obvious that the Commodore does not give a shit about Jimmy. Naturally, this would mean that the Commodore was well into his 50s when he hung out with a 13 or 14 year old girl.
 * As of the penultimate episode of the season, the Commodore has been

Nucky is Tony Soprano's Great Grandfather
Conjoining with the top WMG Theory, All of HB Os Mafia themed series (past, current, and future) might be connected so Nucky might (if not his direct Great Grandfather) be related to the Soprano family in someway or another.

Jimmy is a Sacrificial Lamb.
We're up to episode three of the series, and Jimmy already has this going against him:
 * The prohibition agent knows that Jimmy pulled off the hijacking of Rothstein's delivery.
 * Because of Jimmy's involvement with the above, Rothstein wants him dead.
 * Nucky fired him and strongly suggested he disappear for a while.

Whoever can survive those odds in a show like this is damn near a Determinator, and Jimmy's not likable enough to be that.
 * Well, it seems so, as Jimmy then went to Capone, who was rising through the ranks of Turrio's outfit in Chicago at the time, and wound up becoming one of the top men in Turrio's fit. And as of the last episode, Nucky invited Jimmy back as a hitman to help deal with his rivals.

A related theory: As of “Two Boats and A Lifeguard,” Jimmy has apparently successfully pulled off his coup of power from Nucky in Atlantic City. He’s also played a pivotal role in uniting the Young Future Famous People (Luciano, Lansky and Capone) to likewise take out the old guard gangsters (Torrio and Rothstein). However, Jimmy’s claim to the throne is already shaky, with people pressuring and second-guessing him, Manny Horvitz demanding money and Jimmy being generally volatile as a result of his falling out with Nucky. It stands to reason that while he serves as a means to usher in the new criminal regime of the series, he himself will fall hard, fast and soon, especially if Eli’s comments about Nucky being smarter and unforgiving hold true and Nucky’s plans (pretty obviously pretending to accept defeat to get Jimmy & Co. think they’ve won, as well as encouraging Chalky to rile up his black constituents) pay off.
 * As of the season finale, "To the Lost,".

The opening credits take place in the afterlife
Nucky's ghost haunts AC's beach, having to stand and count one bottle for each person's life his bootlegging business destroyed. He finishes it in September 19, 2010 after which he is allowed to go.
 * For proof of this click here.

The final scene of the series
Will show Jimmy's nearly centenarian son in the modern day being assaulted by crooks in an alley and musing about how the gangsters of old might be a bunch of sonofabitches, but sonofabitches with style.
 * No, just no.

The Indiana Jones canon is a highly fictionalized account of Jimmy Darmody's life
Back in the 70s, a young George Lucas met an old man in a bar. The man began to tell him the story of his life, but he changed anything he didn't like ("...and my mother was a Single Mom Strip... No! No! She was a Southern Belle! And my father was a Scottish Medievallist! He looked like Sean Connery! Really!") and exagerated notably his exploits in The Great War and his brief relation with the Chicago Outfit. However, when he was coming to the part when he returned to Joisey, he realized it could get him in trouble and replaced it all with a ridiculously inaccurate career as an Adventurer Archaeologist that fought Those Wacky Nazis. Lucas liked the story so much that he turned it into a movie franchise.
 * This is somewhat compromised after  in "To the Lost", unless the old man was

Mabel was a daughter to Alma Garret, born after she left Deadwood
It makes sense.

Chalky was present when his father was killed
That's why he knows it was six men who did it, no more, no less.

Seeing the completed bookcase made the young Chalky decide to be a carpenter too and asked his father to assist him during his next job. Unfortunately, this turned out to be a set up and his father was hanged. Being only a child, the murderers decided to not kill Chalky too, but they forced him to watch and scarred his face afterwards to "teach him a lesson".
 * So Chalky is Inigo Montoya? Hmm, makes a certain amount of sense.
 * "Ma name is Chalky White. Yo killed ma Daddy, prepare to die".

Jimmy joined the military because of sexual tension with Gillian.
I was thinking about this in light of the protagonist of The Grifters abruptly leaving for that reason. It would kind of make sense for why Jimmy left so suddenly, and why he dropped out of Princeton- it wouldn't really help, since he'd still be fairly close by.
 * Confirmed in "Under God's Power She Flourishes" . It's also indicated that Jimmy was probably close to getting expelled anyway in circumstances directly attributable to Gillian.

Nucky saved Eddie from anti-German violence.
Eddie has Undying Loyalty to Nucky which doesn't seem at all deserved in light of any of Nucky's behavior to him on the show. There has to be a good reason for it, and it would make sense if he hired Eddie prior to World War I and at the time, also protected him from xenophobia. It would also be really in line with Nucky's personality for him to be quite happy to stoke anti-German xenophobia when it was in his interest- i.e. regarding Hans Schroeder.
 * More than a guess, I believe this was more or less said outright in an early episode.

The damage to Richard Harrow's vocal chords makes it hurt to speak.
Richard is so terse because it's painful for him just to say more than a few words. He pauses mid sentence, not out timidity or to clear his throat, but because it hurts too much to keep going. After suspecting this, I wince in sympathy every time he has to say something.
 * Judging from the scars on his neck and that the damage is mostly on one side, it's likely that Richard Harrow suffers from damage to one of his recurrent laryngeal nerves, this creates the hoarseness to his voice, for a similar example see "Godfather" from Generation Kill. This in itself wouldn't be painful, but the other injuries to his face and mouth are likely to hurt badly, plus we also see that he has difficulty swallowing alcohol. Overall not a pleasant experience.

Adrian is the fifth D'Alessio brother
We know that the D'Alessios have an adult non-criminal brother named Adrian. The age gap seems to be most wide between Lucien (fourth of the criminal gang by age) and Sixtus (fifth), and coincidentally Sixtus also means "sixth" in Latin (it was in fact a popular name for the sixth child born in Roman times). So it makes sense that the order is Leo-Ignatius-Teo-Lucien-Adrian-Sixtus-Pius-two male children, thus making the 9 D'Alessio brothers total mentioned.

Giving birth to Jimmy at 14 left Gillian unable to bear more children
Never stated openly, but seems likely considering the fact that she never had more children despite her promiscuous life and that she unsuccessfully spent the better part of the years from 1905 to 1916 trying to get some rich guy to marry her. The only reason she didn't use The Baby Trap was because she couldn't.

Tommy Darmody will be the one to bring down Nucky
Nucky is loosely based on Enoch L. Johnson, who held power in Atlantic City until his imprisonment in 1941. By that year, Tommy Darmody will be 24.