The apply() function in R can apply functions over margins of arrays or matrices. It avoids explicit loops and applies the given function to each row or column or both. Some key advantages of apply() include avoiding explicit loops, ability to apply various functions like mean, median etc, and ability to apply user-defined functions. Similarly, lapply() and sapply() apply a function over the lists or vectors but lapply() returns a list while sapply() simplifies the output if possible. Functions like tapply() and by() are useful when dealing with categorical variables to apply functions across categories. mapply() applies a function to multiple arguments and is useful for multivariate functions.