In computer science, a lookup table (LUT) is an array that replaces runtime computation of a mathematical function with a simpler array indexing operation, in a process termed as direct addressing. The savings in processing time can be significant, because retrieving a value from memory is often faster than carrying out an "expensive" computation or input/output operation. The tables may be precalculated and stored in static program storage, calculated (or "pre-fetched") as part of a program's initialization phase (memoization), or even stored in hardware in application-specific platforms. Lookup tables are also used extensively to validate input values by matching against a list of valid (or invalid) items in an array and, in some programming languages, may include pointer functions (or offsets to labels) to process the matching input. FPGAs also make extensive use of reconfigurable, hardware-implemented, lookup tables to provide programmable hardware functionality. LUTs differ from hash tables in a way that, to retrieve a value ${\displaystyle v}$ with key ${\displaystyle k}$, a hash table would store the value ${\displaystyle v}$ in the slot ${\displaystyle h(k)}$ where ${\displaystyle h}$ is a hash function i.e. ${\displaystyle k}$ is used to compute the slot, while in the case of LUT, the value ${\displaystyle v}$ is stored in slot ${\displaystyle k}$, thus directly addressable.

Before the advent of computers, lookup tables of values were used to speed up hand calculations of complex functions, such as in trigonometry, logarithms, and statistical density functions.

In ancient (499 AD) India, Aryabhata created one of the first sine tables, which he encoded in a Sanskrit-letter-based number system. In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" Modern school children are often taught to memorize "times tables" to avoid calculations of the most commonly used numbers (up to 9 x 9 or 12 x 12).

Early in the history of computers, input/output operations were particularly slow – even in comparison to processor speeds of the time. It made sense to reduce expensive read operations by a form of manual caching by creating either static lookup tables (embedded in the program) or dynamic prefetched arrays to contain only the most commonly occurring data items. Despite the introduction of systemwide caching that now automates this process, application level lookup tables can still improve performance for data items that rarely, if ever, change.

Lookup tables were one of the earliest functionalities implemented in computer spreadsheets, with the initial version of VisiCalc (1979) including a LOOKUP function among its original 20 functions. This has been followed by subsequent spreadsheets, such as Microsoft Excel, and complemented by specialized VLOOKUP and HLOOKUP functions to simplify lookup in a vertical or horizontal table. In Microsoft Excel the XLOOKUP function^{[further explanation needed]} has been rolled out starting 28 August 2019.^{[citation needed]}

Although the performance of a LUT is a guaranteed ${\displaystyle O(1)}$ for a lookup operation, no two entities or values can have the same key ${\displaystyle k}$. When the size of universe ${\displaystyle \cup }$—where the keys are drawn—is large, it might be impractical or impossible to be stored in memory. Hence, in this case, a hash table would be a preferable alternative.

For a trivial hash function lookup, the unsigned raw data value is used directly as an index to a one-dimensional table to extract a result. For small ranges, this can be amongst the fastest lookup, even exceeding binary search speed with zero branches and executing in constant time.

One discrete problem that is expensive to solve on many computers is that of counting the number of bits that are set to 1 in a (binary) number, sometimes called the population function. For example, the decimal number "37" is "00100101" in binary, so it contains three bits that are set to binary "1".