Adabas was around before SQL had established itself as a standard.
It is extremely fast & reliable & has been known to run literally for years at customer sites.

As quadphonic observed, it uses Inverted Lists, which are really just Indices.

In theory, the performance is comparable with, lets say other all-pervasive SQL Databases you may have encountered out there.
In practice, real-life Adabas DB's run much faster, as SQL has some shortcomings:
for example, you can't formally specify which Index you want to use (instead you have to rely on hints).

My experience of Larrys DB (that & Java is what I work with these days) is that queries often can deteriorate into a full table scan, because the optimizer doesn't always get it right & that can be very costly.

With Adabas you have WITH (Index criteria) & WHERE (non-Index) clauses, which empower the expert programmer.
(There is also a special-case for mixing non-Index criteria with WITH, but lets not get into that here!)

You also have 3 major access-paths:
FIND - sort of like SELECT in SQL
READ - similar to SELECT...ORDER BY, but allows Index or Physical order to be specified
HISTOGRAM - like read, but operates only on Indices. Very fast.

For those who need it, there is an SQL-Engine available for Adabas.

Natural, the programming language, is mostly used together with Adabas, but the same queries can be mapped via the Data-Dictionary (PREDICT) to many different underlying DB's (Adabas, Oracle, DB2, IMS/DB (DL1), VSAM, even System-resources like Main-Memory & Job-Queues. The list is long...)

Natural is in use by many Major International corporations & runs on various Hardware, from Mainframes (IBM, Siemens) to Windows.

The name stams from the term "Natural Language", as you might speak, for example:
MOVE A TO B, or MULTIPLY A BY B
Very similar to Cobol, but much, much more powerful.
It is much easier to understand than the OO/Java/C-like object.method(...) * DWORD -> ~^ ! stuff thats prevalent these days.

Natural was probably the first language to make window-technology readily available on the mainframe & is lightyears ahead of Cobol which is the all-pervasive mainframe language in real-world Fortune500 IT.

It would be unfair to compare Natural with PL/SQL.
While Natural is also used to write Adabas Triggers, it is also a full-function language in its own right, not just some old scripting language.

Software AG also offer various middleware to connect old & new, for example an ILOG JView Java Frontend connected via their EntireX middleware to the old host-based applications is used to allocate Cargo-loks by the german state railways (DB). Through their aquisition of WebMethods, they are also well positioned in the SOA marketplace.