Every computable function can be defined in the Lambda CalculusNamely every function which can ever be computed algoritmically can be expressed in the Lambda Calculus. This statement cannot be proved formally, of course, because the notion of "algoritmic computability" is not defined formally. But, as a matter of fact, this "conjecture" has never been "disproved": not even the most sophisticated modern machines and programming languages can define more functions than those defined by the lambda calculus (or any of the other equivalent formalisms: Recursive functions, Turing machines, etc.).
Example (assuming a language enriched with numerical constant and primitive operations): (\x. x+1) represents the function which, on input x, outputs x+1.
Variables: V ::= x | y | z | ...
Lambda-Terms: M ::= V | (\V M) | (M M)
M1M2 ... Mk stands for ( ... (M1M2) ... Mk)
\x1x2 ... xk.M stands for (\x1(\x2 ... (\xk M) ... ))