Compound Poisson processes are the textbook example of pure jump stochastic processes and the building blocks of Lévy processes. They have three defining parameters: the distribution of the jumps, the intensity driving the frequency at which these occur, and the drift. They are used in numerous applications and, hence, statistical inference on them is of great interest. In particular, nonparametric estimation is increasingly popular for its generality and reduction of misspecification issues. In many applications, the underlying process is not observed directly but at discrete times. Therefore, important information is missed between observations and we face a (non-linear) inverse problem. Using the intimate relationship between Lévy processes and infinite divisible distributions, we construct new estimators of the jump distribution and of the so-called Lévy distribution. Under mild assumptions, we prove Donsker theorems for both (i.e. functional central limit theorems with the uniform norm) and identify the limiting Gaussian processes. This allows us to conclude that our estimators are efficient, or optimal from an information theory point of view, and to give new insight into the topic of efficiency in this and related problems. We allow the jump distribution to potentially have a discrete component and include a novel way of estimating the mass function using a kernel estimator. We also construct new estimators of the intensity and of the drift, and show joint asymptotic normality of all the estimators. Many relevant inference procedures are derived, including confidence regions, goodness-of-fit tests, two-sample tests and tests for the presence of discrete and absolutely continuous jump components. In related literature, two apparently different approaches have been taken: a natural direct approach, and the spectral approach we use. We show that these are formally equivalent and that the existing estimators are very close relatives of each other. However, those from the first approach can only be used in small compact intervals in the positive real line whilst ours work on the whole real line and, furthermore, are the first to be efficient. We describe how the former can attain efficiency and propose several open problems not yet identified in the field. We also include an exhaustive simulation study of our and other estimators in which we illustrate their behaviour in a number of realistic situations and their suitability for each of them. This type of study cannot be found in existing literature and provides several insights not yet pointed out and solid understanding of the practical side of the problem on which real-data studies can be based. The implementation of all the estimators is discussed in detail and practical recommendations are given.