Method of moments when the distribution is unknown. I get that method of moments (MoM) can be used to get the parameters of a specified distribution (under certain conditions). When the distribution is unknown, is it possible to use MoM on some kind of generalized distribution to narrow down to a particular distribution? Like generalized Gaussian, for instance. I am looking for something more general than Gaussians. 
Or is it that having a knowledge of a particular distribution is a pre-requisite for using MoM?
Thanks!
 A: You cannot use the classical 'method of moments' method to estimate
parameters of a distribution without knowing the distribution and
its parameters. 
Examples. For data $X_1, X_2, \dots, X_n$ from a normal population the method of
moments estimator (MME) of the population mean $\mu$ is the sample mean $\bar X$
and the population variance $S^2$ is the MME of the sample variance $\sigma^2.$
By contrast, for data from an exponential distribution, the population mean $\mu$ and
SD $\sigma$ are the same, and so the MME of both $\mu$ and $\sigma$ would be
$\bar X.$ (You could use $S$ to estimate $\sigma$ but it wouldn't be as good.)
Moreover, for data from a population distributed $\mathsf{Unif}(0, \theta)$,
the population mean $\mu = \theta/2$ so the MME of $\theta$ is $2\bar X.$
(However, the maximum likelihood estimator, based on the maximum observation,
is better according to some important criteria for 'goodness' of estimators.)
You could use $S^2$ to estimate $\sigma^2 = \theta^2/12$, but this is not
a particularly good estimator (and not an MME according to the usual
definition).
In summary, it really helps to know what the population parameters are, if you are going to estimate them using the method of moments (or the method of maximum likelihood).
Nonparametric testing and estimation. However, without detailed knowledge of the population distribution
you might use a sample mean, median, standard deviation, upper quartile, etc. to get a
point estimate of the population mean, median, standard deviation, upper quartile, etc. (There might be better estimators, but you would have no way
to know.)
You can also use modern computer intensive methods such as bootstrapping
and (simulated) permutation tests to find confidence intervals for
the population mean, median, or standard deviation, and to test
hypotheses about these population characteristics. 
Because these methods
do not require you to know the population distribution or its parameters,
these methods are often called nonparametric. Examples of traditional
nonparametric methods, based on ranks, to estimate or test hypotheses
about population medians are the Wilcoxon signed rank (one-sample) and
rank sum (two-sample) procedures.
A: Let me add a tangential remark. I agree with the accepted answer that the method of moments is used in the case when one wants to estimate the parameters of a known, parameterised distribution. It is done by analytically calculating moments in terms of the parameters, then expressing the parameters from these equations in terms of the moments in closed form, and substituting the empirical moments to compute the parameter estimates.
However, there is a related option for the case when an explicit, analytical expression is not available but samples can be exactly simulated from the parameterised distribution (for any parameter vector). This is the method of simulated moments.
In the method of simulated moments, we draw $S$ independent samples of size $n$ from the distribution for every proposed parameter vector, repeat with different parameter vectors, and choose the parameter vector where the moments of the data and the moments of the simulated data averaged over the $S$ samples match most closely. I believe Section 2 of my paper summarises the method well.
Literature:


*

*Christian Gouriéroux and Alain Monfort. Simulation-based econometric methods. Chapter 2. Oxford University Press, Oxford, UK, 2002.

*Christian Gouriéroux and Alain Monfort. Simulation based inference in models with heterogeneity. Annales d'Économie et de Statistique, 20–21:69–107, 1991.

