Let $X$ be an algebraic variety over a field $k$ of characteristic not 2. A quadratic space on $X$ is a locally free sheaf ${\scr E}$ on $X$ together with a self dual isomorphism $q\colon {\scr E}\to {\scr E}^\vee$. The study of quadratic spaces over algebraic varieties comes under two broad heads: the non-stable theory and the stable theory via Witt groups.
The non-stable study acquired an impetus with the solution by Quillen and Suslin of Serre's problem on the triviality of algebraic vector bundles on the affine space. In constrast to the linear case, explicit non-trivial quadratic spaces of rank 4 were constructed over ${\bf A}^2_\pi$. Ojanguren and independently Suslin-Kopeiko showed that any isotropic quadratic space over ${\bf A}^n_k$ is extended from $k$. Hence the obstruction to quadratic spaces over ${\bf A}^2_k$ being extended from $k$ lies in the existence of anisotropic quadratic spaces over $k$. In fact it was proved that if $k$ admits an anisotropic space $q_0$ of rank at least 3, then indecomposable quadratic spaces can be constructed over ${\bf A}^2_k$, whose fibre at any rational point is $q_0$. A general theorem, extending these results, has been proved by Raghunathan, for principal $G$-bundles over ${\bf A}^2_k$, where $G$ is a connected reductive group over $k$.
An offshoot of the discovery of non-trivial quadratic spaces over ${\bf A}^2_k$ was a systematic study due to Knus-Ojanguren-Parimala-Sridharan of low rank quadratic spaces over arbitrary commutative rings, via Clifford algebras. This study has led to a better understanding of certain classical questions. We cite an instance: If $D$ is a central division algebra of dimension 16 over $k$, which admits an involution, a classical theorem of Albert asserts that $D$ is a tensor product of two quaternion sub-algebras. A natural question was to determine when an involution $\sigma$ on $D$ also decomposes. Examples of indecomposable involutions were constructed by Amitsur-Rowen-Tignol. A theorem of Knus-Parimala-Sridharan pins down the obstruction to the decomposability of $\sigma$ to the non-vanishing of a certain invariant which they call the Pfaffian discriminant of the involution (the origin of this invariant can be traced back to the work of Jacobson and Tits). Subsequently, the determination of possible discriminants of involutions on a given algebra $D$, $[D\colon k]\geq 16$ become a problem of wider interest. It was not clear a priori whether one could construct on a given algebra $D$ an involution which has trivial discriminant or for that matter one which has non-trivial discriminant. Parimala-Sridharan-Suresh showed that the set of discriminants of involutions on $D$ coincides with the group of reduced norms on $D$ modulo squares. This answers completely the above mentioned question.
We next mention some results in the stable theory initiated by Knebusch, who defined the Witt group $W(X)$ of a variety $X$. The theory is especially interesting for real varieties, where there is subtle interplay between the geometry of $X$ and the real topology of $X({\bf R})$. In this connection, Knebusch raised the question of finite generation of $W(X)$ for smooth real varieties. Finite generation of $W(X)$ was proved by Knebusch himself for $\dim X=1$ and by Ayoub for $\dim X=2$. However, for a smooth affine 3-fold $X$, it was shown that the finite generation of $W(X)$ is equivalent ot the finiteness of $CH^2(X)/2$, where $CH^2$ denotes the group of codimension 2 cycles modulo rational equivalence. (This is indeed the case if $CH^2(X_{\bf C})/2$ is finite. We note in parenthesis that finiteness of $CH^2(Y)/2$ for smooth 3-folds $Y$ over ${\bf C}$ is in general an open question.) A crucial result used to prove this theorem is the finiteness of certain unramified cohomology groups of smooth varieties over ${\bf R}$ proved by Colliot-Thélène and Parimala. More precisely, it was shown that if $n>\dim X$, the group of sections of the Zariski sheaf associated to the presheaf $U\mapsto H^n_{\text{\'et}}(Z,{\bf Z}/2)$ is isomorphic to $({\bf Z}/2)^s$, where $s$ denotes the number of real components of the topological space $X({\bf R})$. This could be thought of as a higher dimensional analogue of a classical theorem of Witt for curves. (There is a recent refinement of this result by Colliot-Thélène and Scheiderer.)
The obstruction to the finite generation of $W(X)$ mentioned above vanishes if the "cycle map" $${\rm cl}\colon CH^2(X)/2\to H^4_{\text{\'et}}(X,{\bf Z}/2)$$ is injective. It was shown by Colliot-Thélène and Parimala that if $X$ is a smooth projective surface over ${\bf R}$, this map is indeed injective. The question of injectivity of this map for aribtrary smooth projective surfaces with rational points has been of general interest. Parimala-Suresh have recently constructed as example of a smooth projective surface (in fact a smooth conic fibration over a curve) with a rational point over a $p$-adic field for which the cycle map is not injective.