Understanding Monte Carlo sampling In rejection sampling or Markov chain Monte Carlo methods, we usually have a target distribution $p(x)$ whose form makes it difficult or impossible to draw samples directly, but we can evaluate $p(x)$ up to a normalising constant. We then sample from a simpler distribution which is proportional to $p(x)$. 
My problem appears here: In order to accept/reject the proposed value $x'$, we evaluate the value of $p(x')$ and see if its within its bounds. 
Why can we evaluate a value $p(x)$, but not sample directly from $p(.)$? It seems like if we can resolve $p(x)$ for all $x$, say uniformly sampled over the support of $p(.)$, we can directly assess the shape of $p(x)$ without any distribution convergence steps. 
If someone could spot where I am confused, it would be very appreciated, as I don´t seem to find any clear explanation on this issue anywhere (must be really trivial..! )
 A: I think what you have in mind is to evaluate $p(x), \forall x \in \Omega$ and then treat it as a discrete distribution and pick an outcome at random and according to the probabilities (which we know how to do for a discrete distribution).
The problem with that, however, is that $|\Omega|$ is often extremely large and in some cases infinite (if $x$ is continuous) making the above approach not practical.
Also, it is important to realize the distinction between being able to draw a sample from a distribution and being able to evaluate a distribution at an arbitrary outcome. The latter does not mean that you can sample from the distribution easily. As far as I know, we only know how to draw samples from a uniform distribution, and sampling from all other distributions (e.g. Gaussian, categorical, etc.) is done by applying various tricks on uniformly drawn samples (I might be wrong on this though).
A: The method is extensively used to sample from a posterior distribution in Bayesian statistics. The following refers to that situation.
Following Bayes' theorem, the posterior distribution is proportional to the prior times the likelihood. In notations:
$$
p(\theta | x) \propto p(\theta) \times p(x | \theta)
$$
Sometimes, $p(\theta | x)$ corresponds to a known distribution, like Student. In that case, values can directly be sampled from that distribution and MCMC is not needed. 
But, $p(\theta | x)$ does not always correspond to a known distribution...
Given the prior and the likelihood, however, you can always resolve the posterior at every parameter value up to a multiplicative constant (cf. Bayes' theorem above). As explained by @Sobi, "Uniform sampling" (as you suggest) will not work for continuous distributions. But, the MCMC method applies.
