  We consider several types of internal queries, that is, questions about
fragments of a given text $T$ specified in constant space by their locations in
$T$. Our main result is an optimal data structure for Internal Pattern Matching
(IPM) queries which, given two fragments $x$ and $y$, ask for a representation
of all fragments contained in $y$ and matching $x$ exactly; this problem can be
viewed as an internal version of the Exact Pattern Matching problem. Our data
structure answers IPM queries in time proportional to the quotient $|y|/|x|$ of
fragments' lengths, which is required due to the information content of the
output. If $T$ is a text of length $n$ over an integer alphabet of size
$\sigma$, then our data structure occupies $O(n/ \log_\sigma n)$ machine words
(that is, $O(n\log \sigma)$ bits) and admits an $O(n/ \log_\sigma n)$-time
construction algorithm.
  We show the applicability of IPM queries for answering internal queries
corresponding to other classic string processing problems. Among others, we
derive optimal data structures reporting the periods of a fragment and testing
the cyclic equivalence of two fragments. IPM queries have already found
numerous further applications, following the path paved by the classic Longest
Common Extension (LCE) queries of Landau and Vishkin (JCSS, 1988). In
particular, IPM queries have been implemented in grammar-compressed and dynamic
settings and, along with LCE queries, constitute elementary operations of the
PILLAR model, developed by Charalampopoulos, Kociumaka, and Wellnitz (FOCS
2020).
  On the way to our main result, we provide a novel construction of string
synchronizing sets of Kempa and Kociumaka (STOC 2019). Our method, based on a
new restricted version of the recompression technique of Je\.z (J. ACM, 2016),
yields a hierarchy of $O(\log n)$ string synchronizing sets covering the whole
spectrum of fragments' lengths.
