Practical non-Turing-complete languages?

Don't listen to the naysayers. There are very good reasons one might prefer a non-Turing complete language in some contexts, if you want to guarantee termination, or simplify code, for example by removing the possibility of runtime errors. Sometimes, just ignoring things may not be sufficient.

The paper Total Functional Programming argues more or less persuasively that in fact we should almost always prefer such a restricted language because the compiler's guarantees are so much stronger. Being able to prove a program halts can be significant in and of itself, but really this is the product of the much easier reasoning that the simpler languages afford. As one component in a hierarchy of languages of varying capability, the range of utility of non-universal languages is quite broad.

Another system that addresses this layering concept much more fully is Hume. The Hume Report gives a full description of the system and its five layers of progressively more complete, and progressively less safe, languages.

And finally, don't forget Charity. It's a bit abstract, but it is also a very interesting approach to a useful but not universal programming language, which is based very directly on concepts from category theory.

BlooP (short for Bounded loop) is an interesting non-Turing-complete language. It's a essentially a Turing-complete language, with one (major) caveat: every loop must contain a bound on the number of iterations. Infinite loops are not allowed. As a result, the Halting Problem can be solved for BlooP programs.