There are certain things BASH is not very good at. There are certain tasks you shouldn't do in bash, unless you really, truly have to. It's often better to switch to a different language for most of these tasks.
Speed. Do we really have to say it? Bash is slow. If speed is an important consideration, then Bash may not be the best choice.
Floating point math. Bash has only integer math. Use bc(1) or AWK instead.
Data structures. Bash does not have Pascal-style records (C-style structs); nor does it have pointers. Any attempt to create advanced data structures (stacks, queues, linked lists, binary trees...) will have to be done with extremely primitive hacks.
Fancy ProcessManagement. Bash has nothing analogous to select(2) or poll(2). Use C instead.
XML and HTML (or alike) parsing. You need external tools or libraries to do this correctly. Use xslt, tidy, xmlstarlet, perl, or some other suitable tool.
Binary data. Bash has no way to store the NUL byte in a variable, so binary data either has to be encoded (and decoded), or kept in a file. You also can't pass the NUL byte as an argument to a program, because the kernel uses C strings for those. Parsing binary data from a file is also a nontrivial problem. Try perl or C instead.
Database queries. When retrieving a tuple from a relational database, there is no way for Bash to understand where one element of the tuple ends and the next begins. In general, Bash is not suited to any sort of data retrieval that extracts multiple data values in a single operation, unless there is a clearly defined delimiter between fields. For database queries (SQL or otherwise), switch to a language that supports the database's query API.
Variable typing. Like most scripting languages, Bash does not really support strong variable types. Variables are loosely categorized as scalar or array, with partial support for an integer type. But really, everything is a string.
Dropping permissions. It can be tough to make a bash script safe to execute as root. In languages like C, perl, and python, you can easily drop privileges at a certain point. With bash, this is tricky, because while you can run su or sudo, you lose variables, and even the executing environment. Use a proper programming language if you have security worries.
Try/catch. Some programming languages let you wrap a command in a try ... catch block. This will interpret the command in a sort of "sandbox", where errors that would normally cause an abort are "caught", and trigger some sort of error-handling code. Bash does not have anything analogous to this. Any bash code you run is real code.
Functions. Bash's "functions" have several issues:
Code reusability: Bash functions don't return anything; they only produce output streams. Every reasonable method of capturing that stream and either assigning it to a variable or passing it as an argument requires a SubShell, which breaks all assignments to outer scopes. (See also BashFAQ/084 for tricks to retrieve results from a function.) Thus, libraries of reusable functions are not feasible, as you can't ask a function to store its results in a variable whose name is passed as an argument (except by performing eval backflips).
Scope: Bash has a simple system of local scope which roughly resembles "dynamic scope" (e.g. Javascript, elisp). Functions see the locals of their callers (like Python's "nonlocal" keyword), but can't access a caller's positional parameters (except through BASH_ARGV if extdebug is enabled). Reusable functions can't be guaranteed free of namespace collisions unless you resort to weird naming rules to make conflicts sufficiently unlikely. This is particularly a problem if implementing functions that expect to be acting upon variable names from frame n-3 which may have been overwritten by your reusable function at n-2. Ksh93 can use the more common lexical scope rules by declaring functions with the "function name { ... }" syntax (Bash can't, but supports this syntax anyway).
Closures: In Bash, functions themselves are always global (have "file scope"), so no closures. Function definitions may be nested, but these are not closures, though they look very much the same. Functions are not "passable" (first-class), and there are no anonymous functions (lambdas). In fact, nothing is "passable", especially not arrays. Bash uses strictly call-by-value semantics (magic alias hack excepted).
- There are many more complications involving: subshells; exported functions; "function collapsing" (functions that define or redefine other functions or themselves); traps (and their inheritance); and the way functions interact with stdio. Don't bite the newbie for not understanding all this. Shell functions are totally f***ed.
Sorting. Bash can't sort data sets. If you need to sort an array, you can either write your own sorting algorithm in pure bash, or you can serialize the data set, pipe it to sort, and then parse it back in. Either way is painful, particularly if your sort doesn't have -z.
On top of these, BASH is not ideal for large programs. If your program is going to be responsible for a lot of tasks, especially interactively, then you may want to consider another interpreter or switch to a compiled language altogether. Large BASH scripts very quickly get in trouble because BASH is slow at a lot of things other interpreters are fast at. Large chunks of BASH code quickly become non-transparent with few ways other than functions to bring structure to your code. BASH scripts are nearly untestable. Even the most purist of bash programmers (and there aren't many!) write code that, when it all adds up, becomes difficult to maintain. BASH has almost no concept of code safety which lets sneaky little bugs crawl in really easily without warning or notice. And when things go wrong (and things will go wrong), really large scripts are very difficult to debug.
If you do plan to write large BASH scripts, make sure to pay even more attention than normal to every single good practice rule and uphold a consistent style throughout the entire code to avoid too much headache later on.