higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
In noncommutative algebraic geometry one represents a scheme by an abelian category of quasicoherent sheaves on the scheme, and looks at more general abelian categories as being categories of quasicoherent sheaves on a noncommutative space.
In derived (higher) noncommutative (algebraic) geometry one instead considers the derived category of quasicoherent sheaves, or more precisely its dg-enhancement or A-infinity-enhancement; dg-enhancements for the derived categories of quasiprojective smooth varieties are essentially unique by the results of Lunts and Orlov. Taking the derived category instead of the abelian loses a bit of information but sometimes the information is sufficient.
In general one represents complex noncommutative spaces by pretriangulated dg-categories. They may be viewed as models for stable $(\infty,1)$-categories. Note that accessible stable $(\infty,1)$-categories are quite close to Grothendieck $(\infty,1)$-topoi; more flexibility one gets from pretriangulated $A_\infty$-categories or, even better, certain class of spectral categories.
This is well into homotopy theory area. Quillen model category structures and homotopy limits in the context of dg-categories were studied by a number of people (including the impressive thesis by Tabuada). On the other hand, over a mixed characteristics, the meaning of such representations is less well understood.
Derived noncommutative geometry has been informally introduced by Kapranov-Bondal and later Orlov around 1990; full framework belongs to Kontsevich, Lunts, van den Bergh, Katzarkov, Kuznetsov, Kaledin. Some of the works of Toen, Vaquie, Keller, Cisinski, Tabuada are properly in this area as well.
In Katzarkov-Kontsevich-Pantev the following definition is given.
A graded complex nc-space is a $\mathbb{C}$-linear differential graded category $C$ which is homotopy complete and cocomplete (has all homotopy limits and colimits).
The derived categories of quasicoherent sheaves on a scheme over $Spec(\mathbb{C})$ is one of the examples; another example is the category of dg-modules over a fixed dg-algebra $A$, which are such that $A$ admits an exhaustive filtration such that the associated graded is a sum of shifts of $A$. Call that category $A$-$Mod$.
Kontsevich calls a complex differential $\mathbb{Z}/2$-graded algebra
smooth if $A$ is a perfect object in the category of $A$-$A$-dg-bimodules (perfect object here means that $Hom(A,-)$ preserves small homotopy colimits);
compact if the total complex dimension of its cohomology $H^\bullet(A,d_A)$ is finite
The category $A$-$Mod$ is a smooth (resp. compact) nc space if the underlying dg-algebras $A$ is; this notion depends on the category and not on the underlying dg-algebra.
The above definition implies that a category $C$ which represents a nc-space in the sense above is triangulated and Karoubi closed. Sometimes this are requirements in another variant of the definition.
A noncommutative space $X$ is a small triangulated category $C_X$ which is Karoubi closed (=every idempotent is a split idempotent) and appropriately enriched over either
spectra $Hom_{C_X}(E,F[i])=\pi_{-i}\mathbf{Hom}_{C_X}(E,F)$
complexes of $k$-vector spaces (i.e. is a dg-category): $Hom(E,F[i])= H^i(\mathbf{Hom}_{C_X}(E,F))$. $C_X$ is $k$-linear over a field $k$ and one writes $X/k$. If instead $k$ is replaced by a ring $R$ then one enriches over complexes of $R$-modules which are cofibrant.
In the early works of the Moscow school (Kapranov, Bondal, Orlov, Kontsevich) one replaces a variety by the derived category of coherent sheaves (or quasicoherent sheaves on that variety, or dg-category (or A-infinity category) enhancements thereof. There are also noncommutative deformations of such derived categories and analogues like the categories corresponding to the so-called Landau-Ginzburg models. Therefore noncommutative derived algebraic geometry (and even noncommutative motives).
Notice that the derived category of coherent sheaves on a variety does not remember all the structure of the original variety hence derived geometry loses often some information (sometimes not); thus derived algebraic geometry is sometimes easier than nonderived.
and a bit earlier treatise on formal (infinitesimal in the sense of formal schemes) aspect as used in the deformation theory is in
The relations to tropical and symplectic geometry are in recent Kontsevich’s talk at 2009 Arbeitstagung:
Homological mirror symmetry is one of the main motivations and statements of the derived noncommutative algebraic geometry
Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel, 1995.
Maxim Kontsevich, Yan Soibelman, Homological mirror symmetry and torus fibrations, in Symplectic geometry and mirror symmetry (Seoul, 2000), 203–263, World Sci. Publ., River Edge, NJ, 2001.
Dmitri Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, arXiv.