I had the following on a daily calendar I use, but I arrived at my answer differently than they did:
I’m just curious to see if the rest of you come up with their answer or mine.
I get 91. The numbers in the sequence go up by 18.
Same answer as Stainz, different reasoning.
Each pair of numbers adds up to 10
I should have said, I want to see if you come up with the same rationale that they used.
At any rate, Stainz, you came about it the same way I did. However, their explanation…
…was much simpler: “Two series. 1, 3, 5, 7, 9 and the same sequence reversed.” I didn’t notice that pattern at all until reading their solution.
The first thing I thought of was…a palindrome. The same sequence backwards and forwards, the last number has to be 1.
Huh…didn’t even think of that one, although that’s closer to what the calendar’s answer is.
Robot Arm, apparently you should be authoring these puzzles. 
I get 91, because the first difference is constant (basically the same as what Stainz said, but with extra jargon).
Without looking at the other answers:
the next number is 1. The first number is decreasing by 2 and the second number is increasing by 2. You could do a more complicated solution, like the first number increases by 2 and the first plus the second number = 10.
Hey, I got the right answer and a unique solution! Yay me! I never get these.
Asimovian, I solved it the same way you did. I think your strategy is the first one I try with any sequence puzzle.
Daniel
i got it by doubling the number, then adding the previous. eg 73 = 2 x 55 - 37
Wow…that’s a different one – I like it! 
How would you get the second number in the sequence, though? In other words, 37 = 2 x 19 - ??
without looking at anybody else’s reply, the final number is 91, which makes the series a palindrome.
I like the cut of your jib, pat.
I also jumped to the palindrome solution. Normally I’ll first check first and second differences for these, but this one was so obviously a palindrome.
I got the same answer as everyone else, but with different reasoning.
I couldn’t help but notice that the first four terms are the values of
n^5 - 15n^4 + 85n^3 - 225n^2 + 292n - 199
for n = 1, 2, 3, 4. Evaluating at n=5 gives 91.
My first thought was the same as Robot Arm’s. After looking for an alternative solution, I came up with Stainz’s.
But before any of that, I was thrown off a little because, technically, what we’re finding is not a number but a digit. If we really were looking for the next number in the sequence 19 - 37 - 55 - 73 - 9, some of these solutions (like Stainz’s), won’t work.
I also figured that the first four numbers were
1x18+1
2x18+1
3x18+1
4x18+1
and the next one should be
5x18+1
but like Thudlow, I missed that the question was asking for a digit not a new number.
I also missed
the palindrome pattern
Did anyone come up with a different answer?
I got to the 9? at the end and thought, huh? But I did the same as other people. I found the constant difference to be 18. Then I noticed that it was a palindrome.
Cool! This is how I figured it out, too–but you were still first!