Formulas of the Fourier transform family
As a complement to the above answers, we can also classify the transformations based on the orthogonality condition(s) used to derive each pair of transformations.

Fourier Transformation:
$$\frac{1}{2\pi}\int^{+\infty}_{-\infty}e^{ikx}dx = \delta(k)$$

Fourier Series and DTFT:
$$\frac{1}{2\pi}\int^{\pi}_{-\pi}e^{ikx}dx = \delta_{k,0}$$
$$\frac{1}{2\pi}\sum^{+\infty}_{k=-\infty}e^{ikx} = \delta(x),\;\;\;\;k\in Z, x\in (-\pi,\pi)$$
I group Fourier Series and DTFT together, because of the duality. The continuous, periodic function in time domain of Fourier Series, can be seen as the frequency domain function in DTFT. Therefore, there is not much difference between the two, at least on mathematical ground.

DFT:
$$\frac{1}{N}\sum^{N-1}_{n=0}e^{i\frac{2\pi}{N}kn} = \delta_{k,0},\;\;\;\;k=0,1,...N-1$$

Note: Take caution when dealing with the Delta functions. 


Kronecker Delta: It can be treated as a discrete elementary function, and commonly used under a summation.
Dirac Delta: It cannot be derived by elementary function, although many introductory texts use $$\delta(x-a) = \lim_{n\to\infty}\sqrt{\frac{n}{\pi}}e^{-n(x-a)^2}$$ to explain it. And treat it as an ordinary function with a value of $\infty$ at $0$, and a value of $0$ otherwise. Delta function is a distribution and has it meaning definite when used under an integral.