Yes, there are several. Here's a few which I personally care about (described in varying amounts of precision). **This is not meant to be an exhaustive list, and reflects my own biases and interests.**

*I am focusing here on questions which have been open for a long amount of time, rather than questions which have only recently been raised, in the hopes that these are more easily understood.*

**MODEL THEORY**

The compactness and Lowenheim-Skolem theories let us completely classify those sets of cardinalities of models of a first-order theory; that is, sets of the form $$\{\kappa: \exists \mathcal{M}(\vert\mathcal{M}\vert=\kappa, \mathcal{M}\models T)\}.$$ A natural next question is to *count* the number of models of a theory of a given cardinality. For instance, Morley's Theorem shows that if $T$ is a countable first-order theory which has a unique model in *some* uncountable cardinality, then $T$ has a unique model of *every* uncountable cardinality (this is all up to isomorphism of course).

Surprisingly, the countable models are **much harder** to count! Vaught showed that if $T$ is a (countable complete) first-order theory, then - up to isomorphism - $T$ has either $\aleph_0$, $\aleph_1$, or $2^{\aleph_0}$-many countable models. **Vaught's Conjecture** states that we can get rid of the weird middle case: it's either $\aleph_0$ or $2^{\aleph_0}$. In case the continuum hypothesis holds, VC is vacuously true; but in the absence of CH, very little is known. VC is known for certain special kinds of theories (see e.g. Vaught's conjecture for partial orders and http://link.springer.com/article/10.1007%2FBF02760651) and a counterexample to VC is known to have some odd properties, including odd *computability-theoretic* properties (https://math.berkeley.edu/~antonio/papers/VaughtEquiv.pdf), but the conjecture is wide open.

*NOTE: VC can be rephrased as a "countable/perfect" dichotomy, in which case it is* not *trivially true if CH holds and is in fact forcing invariant; see e.g. How do we know if Vaught's Conjecture is Absolute?.*

**PROOF THEORY**

If $T$ is a strong enough reasonable theory, we can define the *proof-theoretic ordinal* of $T$; roughly, how much induction is necessary to prove that $T$ is consistent. For instance, the proof-theoretic ordinal of $PA$ is $$\epsilon_0=\omega^{\omega^{\omega^{...}}}.$$ Proof-theoretic ordinals have been calculated for a variety of systems reaching up to (something around) $\Pi^1_2$-$CA_0$, a reasonably strong fragment of second-order arithmetic which is in turn a very very small part of ZFC. It seems unfair, based on this, to list "**finding the proof-theoretic ordinal of ZFC**" as one of these problems, based on how far away it is; but "find ordinals for stronger theories" is an important program.

*See e.g. Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?.*

**COMPUTABILITY THEORY**

I believe the oldest open problem in computability theory is the **automorphism problem**. In Turing's 1936 paper, he introduced - in addition to the usual Turing machine - the *oracle Turing machine* (or *o-machine*). This is a Turing machine which is equipped with "extra information" in the form of a (fixed arbitrary) infinite binary string. Oracle machines allow us to *compare* the non-computability of sets of natural numbers: we write $A\le_T B$ if an oracle machine equipped with $B$ can compute $A$. This yields a partial ordering $\mathcal{D}$, the **Turing degrees**. Initially the Turing degrees were thought to be structurally simple; for instance, it was conjectured (I believe by Shoenfield) that the partial order is "very homogeneous" (there were many different conjectures).

As it turned out, however, the exact opposite happens: the Turing degrees have surprisingly rich structure. See e.g. http://www.jstor.org/stable/2270693?seq=1#page_scan_tab_contents for an early example of this by Feiner, and http://www.pnas.org/content/76/9/4218.full.pdf for a later one by Shore. Indeed, currently the general belief is that $\mathcal{D}$ is *rigid*, and it has been shown (see e.g. https://math.berkeley.edu/~slaman/papers/IMS_slaman.pdf, Theorem 4.30) that $Aut(\mathcal{D})$ is at most countable. The **automorphism problem** is exactly the question of determining $Aut(\mathcal{D})$; I don't have a reference as to when it was first stated, but I vaguely recall the date 1955.

*We can also ask about "local" degree structures - e.g., the partial order of the c.e. degrees, or the degrees below $0'$ - and there are interesting connections between the local and global pictures.*

Another structural question about the Turing degrees is what sort of natural operations on Turing degrees exist. For instance, there is the Turing jump, and its iterates; but these seem to be the only natural ones. **Martin's conjecture** states that indeed, every "reasonable" increasing function on the Turing degrees is "basically" an iterate of the Turing jump; MC has a few different forms, for instance "all Borel functions . . ." or "In $L(\mathbb{R})$ . . .". See e.g. https://math.berkeley.edu/~slaman/talks/vegas.pdf.

**SET THEORY**

An important theme in set theory is the development of *canonical models* for extensions of ZFC. The first example is Goedel's $L$, which has a number of nice properties: a well-understood structure, a "minimality" property, and a canonical (in particular, foring-invariant) definition. We can ask whether similar models exist for ZFC + large cardinals: e.g. is there a "core" model for ZFC + "There is a measurable cardinal"? This is the **inner model program**, and has been developed extensively. Surprisingly, there is an end in sight: in an appropriate sense, if a canonical inner model for ZFC + "There is a supercompact cardinal" can be constructed, then this inner model will in fact capture all the large cardinal properties of the universe.

*I am breezing past a* truly gargantuan *amount of detail here, but the picture is roughly accurate. See e.g. http://www.math.uni-bonn.de/ag/logik/events/young-set-theory-2011/Slides/Grigor_Sargsyan_slides.pdf for more details, as well as the recent presentation https://www.youtube.com/watch?v=MFDVN7UEUSg&list=PLTn74Qx5mPsQlRpBE5OnxMdN3R1d3DLUO&index=4 by Woodin.*

**SET THEORIES**

When someone says "set theory," they usually mean ZFC-style set theory. But this isn't necessarily so; there are **alternative set theories**. As far as I know, the oldest open *consistency* problem here is whether Quine's NF - an alternative to ZFC - is consistent. Seemingly small variations of NF are known to be consistent, relative to very weak theories, but these proofs dramatically fail to establish the consistency of NF. Recently Gabbay (http://arxiv.org/abs/1406.4060) and Holmes (http://math.boisestate.edu/~holmes/holmes/basicfm.pdf) proposed proofs of Con(NF); my understanding is that Gabbay has withdrawn his proof, and Holmes' proof has not been evaluated by the community (it is quite long and intricate).

**FINITE MODEL THEORY**

For a first-order sentence $\varphi$, let the *spectrum* of $\varphi$ be the set of sizes of finite models of $\varphi$: $$\operatorname{Spec}(\varphi)=\{n: \exists\mathcal{M}(\vert\mathcal{M}\vert=n, \mathcal{M}\models\varphi)\}.$$ We can ask what sets of natural numbers are spectra of sentences; in particular, the **finite spectrum problem** (see the really lovely paper http://www.diku.dk/hjemmesider/ansatte/neil/SpectraSubmitted.pdf) asks whether the complement of a spectrum is also a spectrum. It is known, for example, that the complement of the spectrum of a sentence *not using "$=$"* is a spectrum (http://www.inf.u-szeged.hu/actacybernetica/edb/vol07n2/pdf/Ecsedi-Toth_1985_ActaCybernetica.pdf).

There is a complexity theory connection here: a set is a spectrum iff it is in NEXP. So the finite spectrum problem asks, "Is $NEXP=coNEXP$?"

*We can also ask about spectra for non-first-order sentences.*

**ABSTRACT MODEL THEORY**

Abstract model theory is the study of logics other than first-order. The classic text is "Model-theoretic logics" edited by Barwise and Feferman; see (freely available!) https://projecteuclid.org/euclid.pl/1235417263. The field began (arguably) with Lindstrom's Theorem, which showed that there is no "reasonable" logic stronger than first-order logic which satisfies both the Compactness and Lowenheim-Skolem properties.

Shortly after Lindstrom's result, attention turned towards Craig's interpolation theorem, a powerful result in proof theory (see https://math.stanford.edu/~feferman/papers/Harmonious%20Logic.pdf). Feferman, following Lindstrom, asked whether there is a reasonable logic stonger than first-order which satisfies compactness and the interpolation property. As far as I know, this question - and many weaker versions! - are still completely open.

*I believe this is by far the youngest question in this answer.*

Interpreting Goedel(ed. Juliette Kennedy). $\endgroup$ – Timothy Chow Dec 27 '15 at 19:53