Chapter 0 contains some introductory material. In Chapter I supercompact spaces are introduced, and relationships between such spaces, graphs and interval structures are discussed. Various spaces are shown to be supercompact, among these all compact lattice spaces and all compact tree-like spaces. Several examples are given of compact Hausdorff spaces which are not supercompact. Also regular supercompact spaces (in analogy with regular Wallman spaces) are discussed. Spaces which belong to this class are, e.g., all compact metric spaces, all compact orderable spaces, and all compact tree-like spaces of weight at most $c$.

The chapter closes with a discussion of partial orderings on supercompact spaces.

Chapter II deals with superextensions of a given space $X$. The construction and fundamental properties of $\lambda(X,\scr S)$ are discussed in detail, and also the problem of extending continuous mappings to superextensions. A partial ordering of the set of all superextensions of $X$ is defined in terms of separation of the corresponding subbases. Under suitable restrictions on the subbases, this ordering is similar to the usual ordering of Hausdorff compactifications: $\lambda(X,\scr S)\geq\lambda(X,\scr J)$ if and only if there is a continuous surjection of $\lambda(X,\scr S)$ onto $\lambda(X,\scr T)$ which restricted to $X$ is the identity. Results on connectedness and dimensionality of superextensions are given, in particular for $\lambda X$. Among these are the following. The space $X$ is connected if and only if $\lambda X$ is connected and locally connected. If $X$ is normal, then $\lambda X$ is either zero-dimensional (in case $\beta X$ is zero-dimensional) or infinite-dimensional. Certain subspaces of $\lambda X$ which in some respects behave as the remainder $\beta X-X$ of $\beta X$ are studied. The chapter closes with a discussion of convex sets and hyperspaces.

Chapter III concentrates on infinite-dimensional problems. It starts with a discussion of metrizability and superextensions, which, among other things, shows that every separable not totally disconnected metric space has a superextension which is homeomorphic to the Hilbert cube $Q$. The main result of the chapter is that the superextension $\lambda I$ of the unit interval $I$ is homeomorphic to $Q$. This result is obtained by a series of constructions, using inverse limits and near-homeomorphism techniques. Also, subspaces of superextensions homeomorphic to $Q$ are studied.

Chapter IV deals with Wallman compactifications and GA compactifications of Tichonov spaces. After a brief introduction, some conditions for a Hausdorff compactification $\alpha X$ of $X$ to be Wallman, or regular Wallman, are given. The main result in this direction is that if the set of multiple points is Lindelof semi-stratifiable, then $\alpha X$ is a Wallman compactification (in fact, a $z$-compactification). Some results on locally compact spaces and tree-like spaces are also given, and there is a brief discussion of regular supercompact superextensions. The rest of the chapter is concerned with GA compactifications (which include the Wallman compactifications). A central result is that every compact Hausdorff space of weight at most $c$ is a GA compactification of every dense subspace. In particular, every compactification of a separable space is a GA compactification.

Chapter V gives a survey (without proofs) of results too recent to be incorporated in the main text.

The book contains a large number of instructive examples. Many conjectures and open questions are listed.