Next week or "the" next week?

Next week I’m traveling to Paris.
   or
The next week I'm traveling to Paris.

Are both possible? Are they the same?

"The next week" would just about be possible if you were giving a timetable or a list of activities:

The first week in September I'm going to London, the next week I'm moving on to Paris.

But it doesn't really sound very natural.

There is a simple difference between "next week" and "the next week", but I can only explain it based on something that seems almost too obvious to say: we use the phrase "that day" to refer to any particular day we choose, except this particular day in the present. Any day in the past can be "that day", and any day in the future can be "that day", but the day we are in now is not called "that day" because it has a special name, "today".

And the week that includes "today" also has a special name. We might choose to call any other week "that week", but the week that includes today is "this week" and we do not call it "that week".

The days adjacent to "today" also have special names, "yesterday" and "tomorrow".

And now the part that may not be so obvious: No matter what week in the past or future we choose to call "that week", it was, or will be, followed by "the next week", but "this week" is followed by another week with a special name, "next week", not "the next week".

So both sentences are possible, but the correct choice depends on context. "Next week" is the week immediately following the week that includes "today". The week immediately following any other week is called "the next week".

Now please note that "the next week" can also happen in the past, even though it points to the future. Let's take a look at this example that includes both a conditional and a subjunctive:

If I won the lottery, I'd quit my job the first week, plan an amazing vacation the next week and leave on it the week after that.

Obviously we could have used "the next week" two or more times in the same sentence. I have chosen to use "the week after that" just to show another possibility - one that it is actually quite used, in fact.

If it's just the week after the current one, then it's just "next week", and "the next week" would not be appropriate. However, if you are talking about a week following some future week - or really any future point in time, then "the next week" may be used and simply saying "next week" would not be appropriate.

Two weeks from now we're driving to Bologna to visit my in-laws, and the next week we're going to Bordeaux to spend some time in wine country.

Well, the week of January 4^th won't work for me, but the next week would be okay... alright?

Probably a little more common in this usage is “the week after that”, so you don't actually hear “the next week” said very often. But it can be used and sometimes does get used in the sort of ways mentioned above.

Well, obviously, then, the answer to the immediate question at the beginning of this article is:

Next week I'm traveling to Paris.