"What?" (I hear you say.) "Isn't that a police procedural or something?"
It certainly looks like one, and that's part of its genius. We tend to get so wrapped up in space opera that we think that's what science fiction is supposed to look like. But really, much of that is closer to fantasy -- it's showing a world that is arbitrarily different from our own. The very best science fiction has always been the stuff that takes the modern real world, adds a very limited number of very specific changes, and explores what happens next. That's Person of Interest.
The series certainly starts out looking like a simple episodic heroes-save-people-rah-rah. Our initial protagonists are Harold and John. Harold invented an AI, plugged into the Internet, that sees pretty much everything, and is powerful enough to make generally accurate predictions about what's going to happen next. (The AI is a bit improbably good at predicting human actions; that's the science fiction part, and you need to suspend that particular disbelief. Suffice it to say, there is a brilliant episode in Season 4 that shows what the world looks like from the Machine's point of view, that somehow makes it all feel more realistic.) The government uses this to deal with terrorists and the like, and ignores the threats to ordinary people who are, in the grand scheme of things, irrelevant. So Harold brings John (an ex-military type) on board to figure out what's going on with those "irrelevant" people and generally try to save lives.
There's arc from the beginning, but it starts out fairly obvious -- bad guys in the government, a few secret plots, and so on. I came in late in season one, and found season two fine but not obviously thrilling.
But then -- in season three it started turning into more serious speculative fiction. For instance, it began to explore questions of privacy, and the way that such a Machine can make little mistakes that hurt lives. Not everyone is necessarily happy with this state of affairs.
And then we get to season four, and everything changes. Not to give too many spoilers, suffice it to say the story begins *seriously* looking at the elephant in the room. What really happens if you have a panopticon -- a Machine that can not only see but *predict* most human actions? How powerful would such a Machine actually be? Especially if its opinion of what would be best for everybody doesn't agree with yours?
At this point, after season four, this has become a tale of what it is like when the Singularity is happening all around you, and almost nobody *knows* it's happening aside from you and a few others. On the surface, the world still looks basically pretty normal, but this is now the story of what's happening underneath, and it's the most terrifying thriller I've seen in years. What do you do when there is a malignant god growing all around you, complete with powerful followers who will kill to further its agenda?
It's the best work I've seen from J.J. Abrams, and comes with my highest recommendation. I love Doctor Who and Orphan Black, but by now Person of Interest is IMO just plain better overall. The story is deep, scary and well-told; the direction is solid, as is the acting. (One particular delight is Amy Acker in her best role to date as Root, the self-proclaimed high priestess of the Machine, who is a mix of sympathetic, broken and badass in that way Acker excels at.)
Watch it from the beginning -- while it's a slow build, there is an *enormous* amount of continuity here, and the tension of the later story depends on that build-up. But do watch it...