Here we review dark energy, the component that causes accelerated expansion of the universe. We start by reviewing the history of this fascinating discovery, describing in detail how type Ia supernovae were used to measure the expansion rate and find that the expansion is speeding up. We then outline modern evidence for the existence of dark energy, how dark energy is parametrically described, and what its phenomenological properties are. We review the cosmological-constant problem that encapsulates the tiny size of dark energy relative to expectations from particle physics. Next we introduce physical candidates for dark energy, including scalar fields and modified gravity. We end by explaining the controversial anthropic principle, and describe the possible future expansion histories of the universe dominated by dark energy.

Neutrinos have an important role in cosmology, and here we review them in some detail. We review the fascinating history of how neutrinos were first proposed then detected. We then mathematically describe neutrino oscillations. We describe decoupling of neutrinos from the thermal bath, point out the likely existence of the cosmic neutrino background, and discuss prospects for detecting it directly.

One of the problems with the concept of spacetime is that it is hard for us to actually appreciate the implications of living in a curved spacetime, and the origin of this difficulty is that our local spacetime is essentially flat! Hence, all of our understanding of physics – of 'how things work' – has been built on the basis of perceptions that take place in almost flat spacetime. This chapter will provide a pragmatic approach to the measurement of spacetime by illustrating how it is actually not too difficult to obtain an estimate of local curvature by using simple physical quantities, such as the mass and the size of the object. In this manner, we will be able to appreciate that the curvature on Earth is only a few parts in a billion, hence explaining why we perceive everything in the actual absence of curvature. we will learn how to actually bend spacetime reaching the extreme values that are encountered near a neutron star and a black hole, both of which will be discussed more in detail in the following chapters.

Two scientists more than anyone else have contributed in defining our understanding of gravity: Newton in 1679 and Einstein in 1915. The mathematical frameworks the two have developed and proposed, however, are very different. Newton’s gravity is the one we learn at school and is normally taught at university. It provides a very natural interpretation of what we experience - the apple falls from the tree because the Earth attracts it! Einstein’s gravity is studied only in the most advanced courses at the university and provides a very counterintuitive explanation, requiring the concepts of spacetime and curvature. This chapter will provide a first description of the Einstein equations and, although it will not enter into the mathematical aspects of the equations, it will explain the basic concepts behind them. Acquiring a first qualitative understanding of Einstein equations will be useful to comprehend better the concept of spacetime curvature discussed in Chapter 4.

Gravity attracts – this is such an obvious phenomenon that writing this book was not necessary to stress it. Less obvious is that, even before it appears in the form of physical interaction, gravity attracts our attention and our imagination. As soon as we are born, before developing a conscious relationship with the physical universe, we already know gravity at an instinctive level. For the rest of our lives, it will represent the only one of the four fundamental interactions of which we will have conscious awareness. And from which we will often try to escape.

This chapter explains how the researchers of the Event Horizon Telescope Collaboration were able to obtain the first picture of a black hole through radio-astronomical observations. In particular, we first describe the technological strategies that have been exploited in order to obtain a record-high angular resolution. We will also discuss the theoretical aspects that have allowed the collaboration to model the dynamics of the plasma falling onto the black and to produce a large database of synthetic images potentially describing an accreting supermassive black hole. The chapter reviews how the comparison between the theoretical images and the observations has allowed us to deduce the presence of a supermassive black hole with a mass of 6 billion solar masses in the very heart of the giant galaxy M87. The chapter will also summarise the lessons that have been learnt from this epochal achievement and the questions that are still left unanswered about black holes and gravity in the strongest regimes.

Gravitational waves are simply the solutions of the linearized Einstein equations, that is, the solution of the Einstein equations in weak gravitational fields or, equivalently, in spacetimes that are 'almost flat'. In this respect, they are not very different from the sound waves we hear when we listen to music or from the electromagnetic waves we receive when looking at the screen of our cellphone. However, it is not pressure or electromagnetic fields that are propagated, but rather the very same curvature we have encountered already. Hence, gravitational waves represent the propagation at the speed of light of small ripples in the curvature in spacetime. This chapter will make use of simple mechanical analogues to explain how gravitational waves are produced every time a mass or energy is set in motion, and how the amplitude of gravitational waves produced on Earth can only be extremely small.

Gravity has an irresistible grip on our curiosity and is able to drive our imagination to completely different theoretical spaces. This very fact alone sets gravity aside from all other types of physical interactions we know. Indeed, gravity is the only physical interaction of which we have a conscious experience and this awareness is with us every second of our life. In this book we set out to try to address the question: '…what is gravity and how does gravity actually work?'. This book is meant as a guide in a journey that will take us from our basic understanding of gravity, the one that is deeply coded in our brains even at an instinctive level, to the more physically detailed and yet incorrect description provided by Newton’s theory of gravity. The journey will then lead us to the mathematically beautiful and physically profound description that Einstein has proposed with his 'general theory of relativity', and that is elegantly embodied in his field equations.