Function

In APL syntax, a function (or verb) applies to array arguments to produce an array result (although some APLs allow functions to return other syntactic elements). Unlike many programming languages which use parentheses or brackets for function calls, function application in APL is written simply by writing the function next to its arguments. Functions may be primitive functions, derived functions (including trains), or defined functions. They can be used as operands to operators; in early APLs operands were always functions, but nearly all modern APLs allow array operands, for example to the Power operator.

A key feature distinguishing APL from functional programming languages such as Lisp is that APL functions are not first class objects: they cannot be passed directly as arguments to other functions like arrays. Instead, functions might be considered "second class" objects while operators are "third class" and hyperators, if present, are "fourth class" or more. In a few APLs, most prominently A+, a mechanism is provided to pass functions as arguments, making functions first class if a special function call syntax is used.

Function valence
Functions may take zero, one, or two arguments when applied (to pass more arguments, combine them into a single list argument). Which of these possibilities applies is known as the function's valence: The monadic and dyadic cases are distinguished during execution by checking to see whether there is an array to the function's left, but there is no way to determine from context whether a function should be niladic. It must be a property of the function itself. Thus functions are divided into niladic functions, which are always evaluated immediately with no arguments, and non-niladic functions, which may be monadic or dyadic. Niladic functions lack some of the properties expected of functions—they cannot be passed as operands and cannot be produced as derived functions—so they are often considered to be a special kind of syntactic element rather than a function, and "function" sometimes indicates a non-niladic function.
 * A niladic function takes no arguments: it is evaluated immediately when execution reaches it.
 * A monadic function or prefix function takes one argument on its right.
 * A dyadic function or infix function takes two arguments: a left argument and a right argument.

APL's syntax allows a non-niladic function to be called either monadically or dyadically. However, one or both of these cases may give a SYNTAX ERROR regardless of the argument. In this case the function is called while the last case can be produced, it is unlikely to be useful, and almost never discussed.
 * Monadic if a one-argument call can be valid but a two-argument call cannot
 * Dyadic if only a two-argument call can be valid
 * Ambivalent if both cases are possible
 * no name, if neither case is possible (for instance )

Function definition
Every APL comes with primitive functions, and almost always system functions, which are defined by the language. In order to create other functions, there are three general methods. Every APL supports at least one of these but there is no universally supported method.
 * Tacit programming treats functions as values to be manipulated directly: derived functions can be created by applying operators to operands, and trains are functions formed from multiple functions.
 * Anonymous functions are defined by a series of expressions that make reference to function arguments (dfns use the fixed names  and  ). When created they have no name, but they can be assigned one.
 * Defined functions have a function header with syntax that includes the function's name and reflects how it will be used. This is the only way to create niladic functions.

More recent APLs tend to use the earlier, syntactically simpler, methods on the list above, while older APLs use syntactically more complex ones: in particular, only defined functions were used in almost all APLs until the 1990s. Because they create functions as values with no names, the first two types require function assignment to create named functions (usually considered a requirement to build complex programs). Function assignment became more common in the 1980s, and, due to the popularity of anonymous functions, is present in all new APLs.