Higher Moments, what are they good for? Absolutely nothing?
And now seriously - When I studied the basics of probability theory, and even in more advanced topics (random walks, stochastic processes, etc.), I always felt that the mean and the variance were used extensively (and with good reason). I have wondered what possible uses do higher moments have? 
 A: Perhaps a related question is Why are matrices ubiquitous but hypermatrices rare? in the sense that higher-order (mixed) moments can be regarded as hypermatrices or tensors. For example, the second order moment of a random variable $x\in\mathbb{R}^n$ might be written $\mathbb{E}[x\otimes x]$. The third order moment is $\mathbb{E}[x\otimes x\otimes x]$, and so on.
The (relative) lack of extensive use of higher order moments may be because questions about them inherit the difficulties that come with all tensor problems (Most tensor problems are NP-hard).
A very clear example of higher order moments in use is independent component analysis (ICA), which essentially amounts to a tensor decomposition of the fourth order cumulant (see here, for example). ICA is an instance of a slightly more general problem known as blind source separation, which has wide-ranging applications.
A: If you compute the higher moments for a random variable $X$, and these coincide with the normal distribution, you know that $X$ is also normal.
This can be quite powerful when you want to show that something converges to a normal distribution in the limit. A similar method works for cumulants.
A: I know of two such applications:


*

*The Berry-Esseen theorem: A quantitative version of the central limit theorem which uses the third (centered) moment. 

*Some Orlicz spaces: One can define the Orlicz spaces, which are "$L^p$-esque" spaces. The ones which fit to the function $\Psi_\alpha (x) = exp(x^\alpha)-1$ are supposed to conclude RVs which decay at least as fast as RV whose density is $C\cdot exp(x^{-\alpha})$ (so for $\alpha=2$ one gets sub-gaussian RVs and for $\alpha=1$ one gets "sub-exponential" RVs). The norm in these spaces controls the tails of the RV, compared to a random variable of density $C\cdot exp(x^{-\alpha})$. Then the norm of a RV $X$ is equivalent (up to universal constants) to the quantity
$$
\sup_{p\ge1} {\frac{(\mathbb{E}X^p)^{\frac{1}{p}}}{p^\frac{1}{\alpha}}}
$$
A: I have an overly negative view of this. Higher moments can be useful in some theoretical contexts, but overall they are indeed not used much. This is analogous to the Taylor series of a function. It is rare outside of a few specific areas of math that you ever need anything more than the second order terms in the Taylor series of a function.
