Is there any link between all of the American Horror Story series?

If you pay attention to the timeline of each seasons you will know that Freak Show happened before Asylum and yes, it does explain why Pepper was sent to the Asylum. Sister Mary Eunice did not come to collect Pepper from the Freak Show. Elsa sent Pepper back to Pepper's sister. However, because Pepper was a freak, her sister and the husband frame her for killing their baby (infact, the baby was killed by her sister's husband) and sent Pepper to the Asylum.

Another link is that the Nazi doctor in Asylum was the one chopped Elsa's legs off.

These connections, in fact, are not really important. All the seasons can be watched separately without any difficulties to understand.

There are a few connections between Season 5 Hotel and Season 1 Murder House.

A major connection between the two seasons is when The Countess, from Season 5, visits Dr. Montgomery, the abortion doctor that worked from the Murder House's basement in Season 1, in a flashback to 1925. While this hasn't been fully explained yet as to why The Countess sought out Dr. Montomgery, it is a direct connection between the two seasons.

Some minor connections:

• The realtor, Christine Estabrook Marcy, that sold the Harmons the Murder House in Season 1, sells the Hotel Cortez to a man named Will Drake in Season 5. In a later episode, the realtor is seen staying in a room at the Hotel Cortez.
• The dead roam the halls of the Hotel Cortez much like the souls that are stuck in the Murder House.

Another connection has surfaced recently (spoilers for the S5 season finale):

Sarah Paulson will reprise her Season 1 American Horror Story role during this season’s finale. That’s right, Paulson will be playing two characters this year.

[...]

With the departure of Jessica Lange, Paulson has become American Horror Story’s most regular leading lady, but back in Season 1 her part was relatively tiny. She played Billie Dean Howard, alleged psychic and close friend to Lange’s Constance Langdon, and according to Entertainment Weekly, Billie Dean is coming back. The finale will see Howard scoping out the Hotel Cortez as a subject for her very own reality series featuring her psychic abilities.

American Horror Story - All Seasons Opening Credits (Season 1 to 5)

I love watching the opening credits to all 5 Seasons of American Horror Story.

I analyzed them and there are definitely some similar themes and characters running through each, referencing each different season.

First there is an overall theme of red, black and white. The characters wear black and white, "Coven's" opening sequence is all in black & white, and there is a white dress in many of the opening sequences. There is red, appearing as either as blood or fire. There are children in various forms, signifying birth. Skeletons appear often, signifying death. There are sexual images and women in high heels, signifying fornication. The appearance of children in every opening sequence (with the exception of "Coven" and "Freak Show"), I believe touches on the overall theme of the show, as the beginning of life, creation, and the essence of innocence being corrupted by evil. Each season shows how everyone, no matter how "innocent," has a dark side, some darker than others, and that this is human nature.

There is also a lot of religious imagery, especially in the opening sequences to "Murder House," "Asylum," and "Hotel," which definitely have similar tones. These seasons are also about specific places and the people who inhabit them, a house, an insane asylum, and a hotel. All are places that are haunted by negative energies and dark spirits. All three of these opening sequences have a lot of blood, feature very dark images of people mixed in with images of innocence, like the white dress and the children. "Freak Show" and "Coven" are a little different as they are more about certain "subcultures," such as that of witches and freaks, although "Freak Show" has a lot of red with the clowns, and "Coven" with the fire.

Some specific references to other seasons I noticed were: In "Coven's" opening sequence there is an alien looking creature, which I believe is a tie-in to the "alien" that abducts Kit in "Asylum." There is a carousel in "Freak Show" which references "Hotel," because that's where Holden is abducted from. The children in "Hotel" reference the children in "Murder House." The nun in "Hotel" references "Asylum." So they all tie in somehow.

I also noticed how "Asylum" has the girl walking up the stairs backwards, which references "The Exorcist." And the killer clowns in "Freak Show," reference, Stephen King's "It." "Hotel," with its ten commandments, references "Seven," and the show itself is eerily similar to "The Shining." This all ties in to the overall theme of "American horror stories." Also every show references several different horrific true events in history that have happened, which is probably my favorite part of the show. Life is all one big "Horror Story."

That's all I have so far.