Open questions in "Spin geometry"

Spin geometry is an active field and of course is not exhausted in the book of Lawson and Michelson. In fact, nowadays, there are new books on the topic, including more recent results.

In the introduction of N.Ginoux's book, one finds,

''...one of the most famous achievements of spin geometry was the discovery of a topological obstruction to positive scalar curvature, as a relatively straightforward application of Atiyah-Singer Index theorem''

and below

''...the non commutative geometry made the Dirac operator one of its keystones, as it allows to reconstruct the given Riemannian spin manifold from its canonical spectral triple''

or

''...special eigenvectors of the Dirac operator, called Killing spinors, have become some of the physicists' main tools in the investigation of supersymmetric models in string theory...''

These are just a few reasons that make spin geometry so important.

Nowadays, spin geometry (and all these that it includes), is still very active in several different directions, especially in differential geometry, representation theory, functional analysis, etc. For example, computing the spectrum of the Dirac operator on certain manifolds is a widely open problem (there are a few spaces that we have a complete picture and most of them are homogeneous).

Moreover, (and since the question focus on developments of the spin geometry itself): The last decades there is an enormous progress on Dirac operators related to metric connections different than the Levi-Civita connection and their relationship with the field of non-integrable $G$-structures and special geometries. Such connections are not any more torsion-free (I mean metric connections with skew-torsion, vectorial torsion, etc) and under specific conditions, they become nice replacements of the L-C connection, in the sense that they preserve the special geometric structure (as the L-C connection does in the integrable case). On the other hand, there is a plethora of special structures carrying such connections, e.g. Sasakian, almost Hermitian (e.g. nearly-K\"ahler), or $G_2$-geometries.

In fact, this part of research, i.e. Dirac operators associated to metric connections with skew-torsion, like the characteristic connection $$\nabla^{c}=\nabla^{g}+\frac{1}{2}T$$ associated to a special $G$-structure, where $0\neq T\in\Lambda^{3}T^{*}M$ denotes the torsion of $\nabla^{c}$,
became a subject of interest in theoretical and mathematical physics as well, e.g. in string theory of Type II, where the basic model consists of a Riemannian spin manifold endowed with a 3-form, a dilation function and a spinor field, which is actually parallel under the characteristic connection $\nabla^{c}$ (interpreted as supersymmetric transformation, while the torsion $T$ plays the role of B-field).

From the mathematical point of view, the most famous of such type Dirac operator is the ''cubic Dirac operator'' (with applications both in representation theory and differential geometry). Notice finally that nowadays, there are even generalizations of Killing spinor fields, etc.

As @314159 has explained, Spin geometry on "spin manifolds" is a very active field of research, but the subject goes way beyond the domain spin manifolds. The key point to notice is that every pseudo-Riemannian spin manifold $(M,g)$ admits a bundle of irreducible Clifford modules over the bundle of Clifford algebras $Cl(M,g)$, which is usually called the "spinor bundle", but remarkably enough, the opposite is not true. In other words, there exist manifolds which admit "spinor bundles" but do not admit any spin structure. This is already clear just by looking at the literature on $Spin^{c}$ manifolds, but it turns out the possibilities go beyond $Spin^{c}$ manifolds as well. In fact, so you can see a very recent paper on the subject, the following paper by C. Lazaroiu and Carlos Shahbazi:

https://arxiv.org/abs/1606.07894

classifies all the spinorial structures associated to a bundle of irreducible real Clifford modules, and gives the topological obstructions for the existence of such bundles on any pseudo-Riemannian manifold. You can check that these obstructions are in general different from that required by a spin structure, and correspond to what they call a "Lipschitz structure". They even have some application of his formalism to physics. This proves that "spin geometry" is not always associated to a spin structure and indeed it is a very rich and subtle construction. I think spin geometry in this more general setting is very little explored, and I believe is going to attract a lot of attention in the future.