  Self-assembly is a ubiquitous process in synthetic and biological systems,
broadly defined as the spontaneous organization of multiple subunits (e.g.
macromolecules, particles) into ordered multi-unit structures. The vast
majority of equilibrium assembly processes give rise to two states: one
consisting of dispersed disassociated subunits, and the other, a bulk-condensed
state of unlimited size. This review focuses on the more specialized class of
{\it self-limiting assembly}, which describes equilibrium assembly processes
resulting in finite-size structures. These systems pose a generic and basic
question, how do thermodynamic processes involving non-covalent interactions
between identical subunits ``measure'' and select the size of assembled
structures? In this review, we begin with an introduction to the basic
statistical mechanical framework for assembly thermodynamics, and use this to
highlight the key physical ingredients that ensure equilibrium assembly will
terminate at finite dimensions. Then, we introduce examples of self-limiting
assembly systems, and classify them within this framework based on two broad
categories: {\it self-closing assemblies} and {\it open-boundary assemblies}.
These include well-known cases in biology and synthetic soft matter --
micellization of amphiphiles and shell/tubule formation of tapered subunits --
as well as less widely known classes of assemblies, such as short-range
attractive/long-range repulsive systems and geometrically-frustrated
assemblies. For each of these self-limiting mechanisms, we describe the
physical mechanisms that select equilibrium assembly size, as well as potential
limitations of finite-size selection. Finally, we discuss alternative
mechanisms for finite-size assemblies, and draw contrasts with the size-control
that these can achieve relative to self-limitation in equilibrium,
single-species assemblies.
