From Matthew Belinkie in 2010 -- it's important to understand the Saw series in the context of LOST:

I think to understand the Saw movies, you have to consider another cultural phenomenon that came on the scene in 2004: a little show called Lost. The lesson everyone in Hollywood took from JJ Abrams’s runaway success was that audiences could be enthralled, not repelled, by huge mysteries that unfolded over years. The fans gathered online to obsess over theories and details, the more obscure the better. The Saw producers took this lesson to heart, and built the Saw sequels to be full of twists, complete with Lost-esque flashbacks and lots of loose ends. Want to know what’s in that mysterious wooden box that Jigsaw leaves for his ex-wife? Tune in next year.

At the end of Saw I, Dr. Gordon (Cary Elwes’ character) staggers away, dragging his bloody stump behind him, to bring police to the Bathroom of Doom in time to save Adam, whom he’s just shot (if you haven’t seen the movie, that sentence will seem strange). We know from the discovery of Adam’s rotted body in Saw II that help never arrived. However, we don’t learn what actually happened to Dr. Gordon after he left that room… until Saw VII (akaSaw 3D), when Cary Elwes makes his triumphant return. The series moves forward by looping backwards.

This is the kind of thing that drives Saw fans crazy, and sends them scrambling to their computers to exchange theories. On the House of Jigsaw message board, there are 67,000 posts about Saw V, 81,000 posts about Saw VI, and 66,000 posts aboutSaw 3D, which hasn’t even been released yet. And, taking another page from theLost playback, the actors, writers, and directors of the series drop by the message boards frequently to answer questions and offer teasing nuggets of information.

Just FYI: there are no similar message boards about I Still Know What You Did Last Summer.

Rumor has it that they're working on Saw 8.

Imagine the complexity of storyline they have to take into account:

If you’re a Saw virgin who ignored my warning and kept reading, first of all, don’t try that stuff with Jigsaw. It tends to end poorly. Secondly, you’re probably thinking, “C’mon Matt, they’ve pumped out a Saw sequel every Halloween for the past seven years. The only way you make movies that quickly is if you’re letting the interns write them for college credit. How tightly-plotted could these films possibly be?” Well, here are some examples:

• In Saw III, we see a character read a letter and burst into tears, shortly before flying into a homicidal rage. We don’t learn who wrote this letter until Saw IV. We don’t learn what it said until Saw VI.
• Saw III shows us the five minutes immediately after the end of Saw II.Saw IV shows us what happens immediately after THAT.
• Saw IV actually takes place during the events of Saw III, which is only revealed when a character from Saw IV literally walks into the final scene of Saw III, about two seconds after the previous film cut to black.
• Saw V picks up about thirty seconds after Saw III.
• You see the traps from Saw I being set up in flashback sequences during Saws III and V. You see the traps from Saw II being set up in flashback sequences during Saws III and V. You see the traps from Saw III being set up in flashback sequences during Saws V and VI. And I’m not talking about merely reusing footage – I’m saying that the latter movies recreated earlier sets and brought back actors who were chronologically deceased, to show us new information about things we’d already seen.
• After the credits of Saw VI, you see something that took place between the events of Saws II and III. It may be a critical clue to the denouement ofSaw 3D, or a red herring.
• Did I mention that the main villain, Jigsaw, dies at the end of Saw III, and yet the events of all the subsequent movies are planned by him? And no, there’s nothing supernatural about it.

Matt points out that Fight Club highly influenced Saw, too:

I would bet serious money that one of the inspirations for Saw was Fight Club (1999). At one point, Tyler Durden points a gun at the night clerk of a convenience store, drags him out back, and declares, “Raymond! You’re going to die.” He interrogates poor Raymond and learns that his dream was to become a veterinarian, but it was too hard. “I know where you live,” Tyler says. “If you’re not working to be a veterinarian in six weeks, you will be dead.” He lets Raymond go, and then muses, “Tomorrow will be the most beautiful day of Raymond K Hessel’s life. His breakfast will taste better than any meal you and I have ever tasted.”

Jigsaw is Tyler Durden, without the abs or the sense of humor. He sees his traps as acts of mercy. He doesn’t want to hurt these people. He wants them to stop being crooked cops, loan sharks, philandering husbands, drug dealers, etc. He may put you in a death trap, but he is rooting for you to escape. Jigsaw believes that those who survive will be cured of their vices and redeemed through their suffering. (There’s something very medieval about these purification rituals, and I don’t think it’s a coincidence that he’s often seen in a hooded robe, like a monk.)

Jigsaw sometimes refers to the traps as “games,” but he also calls them “tests,” and the people inside them “subjects.” In other words, he’s performing experiments. He’s an engineer by training, and he sees the people he selects as defective machines in need of repair. Once upon a time, he worked with his ex-wife at a health clinic to heal people the old fashioned way. But fighting addiction and changing patterns of negative behavior is a grueling, uphill battle. John Kramer was determined to find a quick fix for a broken soul.

In other words, Jigsaw’s games are Star Trek solutions to Babylon 5 problems. “You and your wife don’t talk anymore? I bet I can design a machine for that.”

What fascinates me is, even after six films, there is little or no evidence that Jigsaw’s traps actually “fix” anyone. First of all, the number of people who survive is vanishingly small. Here are the movies in which the main person being tested loses their “game”:  Saw I, Saw II, Saw III, Saw IV, and Saw VI. Saw V is the only film in which the some of the characters actually “win” the movie’s central test. And even then, these guys only survive the hard way, through massive bloodshed, as opposed to the easy way, simple teamwork.