As a scientist, one is defined by the kind of problems one works on, the conferences one attends, the journals one publishes in, and the journals one reads. All of these except the last are more or less publicly available information.
Only you know precisely which journals you choose to read, yet they're an essential part of your scientific identity. They determine the kind of new advances you're likely to be aware of and where your research may turn in the future.
I've made my Mendeley library public, so anyone can see in great detail not only what journals I read but which articles I read. But most of you are probably not interested in quite that level of detail, so here's a list of the journals I follow closely. I collect their RSS feeds (with Google Reader) and read at least the title of every article they publish. I've grouped them into 3 main categories, but otherwise they're in no particular order. Those listed in bold are journals where I have published; they also tend to be the journals most heavily represented in my Mendeley library.
Numerical Analysis and scientific computing:
- BIT Numerical Mathematics
- Journal of Scientific Computing
- SIAM Journal on Scientific Computing (SISC)
- SIAM Journal on Numerical Analysis (SINUM)
- Mathematics of Computation
- Numerische Mathematik
- Applied Numerical Mathematics (APNUM)
- Computational Science and Discovery
- Journal of Computational and Applied Mathematics
- Journal of Computational Physics (JCP)
- Computing in Science and Engineering (CiSE)
- International Journal of Numerical Methods in Fluids (IJNMF)
- IMA Journal of Numerical Analysis
- ACM Transactions on Mathematical Software (TOMS)
- Computer Physics Communications
- Acta Numerica
- math.NA on ArXiv
I would add Communications in Computational Physics and Computational Methods in Applied Mathematics, but as far as I know they have no RSS feed.
- Physica D: Nonlinear Phenomena
Here I would add Communications in the Mathematical Sciences, which also has no RSS feed.
General applied math:
- SIAM Review
- SIAM Journal on Applied Math
- IMA Journal of Applied Mathematics
Although ArXiv isn't a journal, I've included it here. Indeed, I find useful articles in that feed much more often than for most of the listed journals.
What does your list look like?