## 1 Introduction

In the 5th century B.C., Democritus postulated the existence of smallest objects that all matter is built from and called them ‘atoms’. In Greek, the prefix ‘a’ means ‘not’ and the word ‘tomos’ means ‘cut’. Thus, atomos or atom means uncuttable or indivisible. According to Democritus’ theory of atomism, “Nothing exists except atoms and empty space, everything else is opinion.” Though variable in shape, Democritus’ atoms were the hypothetical fundamental constituents of matter, the elementary building blocks of all that exists, the smallest possible entities. They were conjectured to be of finite size, but homogeneous and without substructure. They were the first envisioned end of reductionism.2500 years later, we know that Democritus was right in that solids and liquids are composed of smaller entities with universal properties that are called atoms in his honor. But these atoms turned out to be divisible. And stripped of its electrons, the atomic nucleus too was found to be a composite of smaller particles, neutrons and protons. Looking closer still, we have found that even neutrons and protons have a substructure of quarks and gluons. At present, the standard model of particle physics with three generations of quarks and fermions and the vector fields associated to the gauge groups are the most fundamental constituents of matter that we know.

Like a Russian doll, reality has so far revealed one after another layer on smaller and smaller scales. This begs the question: Will we continue to look closer into the structure of matter, and possibly find more layers? Or is there a fundamental limit to this search, a limit beyond which we cannot go? And if so, is this a limit in principle or one in practice?

Any answer to this question has to include not only the structure of matter, but the structure of space and time itself, and therefore it has to include gravity. For one, this is because Democritus’ search for the most fundamental constituents carries over to space and time too. Are space and time fundamental, or are they just good approximations that emerge from a more fundamental concept in the limits that we have tested so far? Is spacetime made of something else? Are there ‘atoms’ of space? And second, testing short distances requires focusing large energies in small volumes, and when energy densities increase, one finally cannot neglect anymore the curvature of the background.

In this review we will study this old question of whether there is a fundamental limit to the resolution of structures beyond which we cannot discover anything more. In Section 3, we will summarize different approaches to this question, and how they connect with our search for a theory of quantum gravity. We will see that almost all such approaches lead us to find that the possible resolution of structures is finite or, more graphically, that nature features a minimal length scale – though we will also see that the expression ‘minimal length scale’ can have different interpretations. While we will not go into many of the details of the presently pursued candidate theories for quantum gravity, we will learn what some of them have to say about the question. After the motivations, we will in Section 4 briefly review some approaches that investigate the consequences of a minimal length scale in quantum mechanics and quantum field theory, models that have flourished into one of the best motivated and best developed areas of the phenomenology of quantum gravity.

In the following, we use the unit convention , so that the Planck length is the inverse of the Planck mass , and Newton’s constant . The signature of the metric is . Small Greek indices run from 0 to 3, large Latin indices from 0 to 4, and small Latin indices from 1 to 3, except for Section 3.2, where small Greek indices run from 0 to , and small Latin indices run from 2 to . An arrow denotes the spatial component of a vector, for example . Bold-faced quantities are tensors in an index-free notation that will be used in the text for better readability, for example . Acronyms and abbreviations can be found in the index.

We begin with a brief historical background.