How closely do Sherlock Episodes relate to the original stories?

I will try to stay away from what Christian has already posted which already is a great answer. I can answer this from a reader's perspective.

Sherlock holmes stories by Doyle can be classified into Long stories and short stories. There are only 4 long stories of which 2 of them are already made into episodes two of them which haven't made it into episodes are the Sign of Four (If rumors are true I think there is a sign of three planned for Season 3 I think thats that adaptation) and Valley of Death (Although I hardly doubt we will see an adaptation of that because this story has a link to Moriarity since he is dead in the series we can put this out of the question). As for the short stories there are 56 short stories which are classified into Adventures, Memoirs, Return and the His Final Bow categories.

The stories the episodes are based off mainly are as follows.

A Study in Pink - A Study in Scarlet (Long story)

The Blind Banker - Mostly based off Five Orange Pips (Short Story)

The Great Game - Mostly based off The Adventure of Bruce Partington Plans (Short Story)

A Scandal in Belgravia - A Scandal in Bohemia (First short story in Adventures of Sherlock Holmes)

Hound of the Baskerville - The Hound of the Baskervilles (Long Story)

The Reichenbach Fall - Final Problem (Last short story of Memoirs of Sherlock Holmes)

To closely relate to the attention to detail given by the creators of the show In the first episode Study in pink when anderson sees the pink lady's body with word Rache written on the floor he surmises that Rache could be a german term referring to Revenge actually but sherlock dismisses him saying that it was indeed intended to be Rachel but in the book its actually the other way, The police start to look for a girl named Rachel and then Sherlock holmes asks them to stop looking for Rachel and says that Rache means German for Revenge.

In the great game too. The plot's core was straight off the short story, in the book it was plans for submarine but in the episode it was a missile program which gets stolen, The same character name Andrew West who dies and the part where the body gets placed near a train tunnel all were right of the short story.

The Hound of the Baskerville was significantly changed from a family plot motive in the book to a biological weapon in the series mainly because creators didn't want to show a hound in the episode they wanted it to be modern so current viewers can relate but the core of the story is driven by the characters created by Doyle.

There were also even other stories referenced in the episodes like the Story of the Geek interpreter (Adventure of the Greek Interpreter), Speckled Blonde (Adventure of the Speckled Band etc). The best part about the show is that it's got equal amount of content for readers and non-readers alike to enjoy about the show. it's not like ones who haven't read the stories won't enjoy the show at all.

These are the specific references I could think of right now. I will update this answer if something else comes to my mind.

They are not merely taking the name. Apart from many smaller nods to the originals (like the title) and little allusions to characteristic conversations from the original stories, there are many bigger story elements from the original that can be found in the episode's story, yet often set into a slightly different context or maybe even parodied.

So the episodes are indeed inspired by the original stories to a large degree, but still tend to differentiate themselves, be it by modernizing certain aspects, parodying things, extending by additional stuff (most of the original stores were just short stories in the end), or just combining elements from the original in interesting and unique ways.

But this is only a very broad answer. Given that I only have rather vague memories to the original stories (and haven't read all of them either), I'll leave actual examples or further explanations to other answers.

In addition to the excellent answers already provided, there essence of the environment created in the new Sherlock series is a modernized, but true to the source mirror of the original.

For example, in the original, Sherlock is manic between cases, and addicted to cocaine and opium. In the modern version, he's a manic between cases, addicted to nicotine patches.

Another example is Watson, and army doctor, injured in the war in India in the original, injured in Afghanistan in the modern version.

In short, though they've updated the elements, I think they've done an excellent job (better than any other attempt) of holding true to the essence of the stories and the characters.

I recently read "A Study in Scarlet" and noticed a number of other similarities to the story:

• Sherlock is introduced to Watson the same way - Stamford, an old medical acquaintance of Watson's, runs into him in London and learns that he is looking for a flatmate; he had earlier spoken to Sherlock and learned that Sherlock was seeking the same thing, so he introduces the two of them.

• At their first introduction, Sherlock deduces that Watson has recently returned from a foreign war.

• As Dredd said, "Rache" was written in blood in the book near one (or both) of the victims, while it's carved into the floor near the fourth victim in the show.

• The killer in both cases (spoiler-free) has the same occupation.

• In both the book and the show, the killer gives his victim a choice between two pills, one of which has no effect and the other of which is poison, and takes the remaining pill himself.

Two additional notices:

• In the story "The Sign of the Four", Watson hands a watch (his late brother's) to Sherlock to see what he can deduce about its owner; one of the things Sherlock says is "Look at the thousands of scratches all round the hole [keyhole for winding the watch]; marks where the key has slipped. What sober man's key could have scored those grooves? But you will never see a drunkard's watch without them." In "A Study in Pink", Sherlock borrows Watson's mobile phone (given to him by his sibling) and makes a similar deduction based on scratches around the charging port.

• In the story "The Hound of the Baskervilles", Sherlock makes a reference to "that little affair of the Vatican cameos"; the phrase "Vatican cameos" is also used by Sherlock in "A Scandal in Belgravia" to warn Watson of the trap hidden in Irene Adler's safe.