The field of succinct data structures has flourished over the last 16 years. Starting from the compressed suffix array (CSA) by Grossi and Vitter (STOC 2000) and the FM-index by Ferragina and Manzini (FOCS 2000), a number of generalizations and applications of string indexes based on the Burrows-Wheeler transform (BWT) have been developed, all taking an amount of space that is close to the input size in bits. In many large-scale applications, the construction of the index and its usage need to be considered as one unit of computation. Efficient string indexing and analysis in small space lies also at the core of a number of primitives in the data-intensive field of high-throughput DNA sequencing. We report the following advances in string indexing and analysis. We show that the BWT of a string $T\in \{1,\ldots,σ\}^n$ can be built in deterministic $O(n)$ time using just $O(n\logσ)$ bits of space, where $σ\leq n$. Within the same time and space budget, we can build an index based on the BWT that allows one to enumerate all the internal nodes of the suffix tree of $T$. Many fundamental string analysis problems can be mapped to such enumeration, and can thus be solved in deterministic $O(n)$ time and in $O(n\logσ)$ bits of space from the input string. We also show how to build many of the existing indexes based on the BWT, such as the CSA, the compressed suffix tree (CST), and the bidirectional BWT index, in randomized $O(n)$ time and in $O(n\logσ)$ bits of space. The previously fastest construction algorithms for BWT, CSA and CST, which used $O(n\logσ)$ bits of space, took $O(n\log{\logσ})$ time for the first two structures, and $O(n\log^εn)$ time for the third, where $ε$ is any positive constant. Contrary to the state of the art, our bidirectional BWT index supports every operation in constant time per element in its output.