How the continuity equation $\partial_\mu j^\mu=0$ means current conservation? I have seen written (for example in Peskin's and Schroeder's QFT book) that $\partial_\mu j^\mu=0$ means that the current $j$ is conserved. I do not understand how this comes to be: I do not understand what "conserved" means in this context: usually I would say that $j$ is conserved if $\frac{dj}{dt}=0$, and this is clearly not what is in stake here.
 A: Let's rewrite that formula a little clearer. I'll (suggestively) write $j=(\rho,\vec{j})$, where $\rho=j^0$ is the 0th component of the 4-vector, and $\vec{j}$ is the spacial component of the 4-vector. I'll call $\rho$ the density, and $\vec j$ the current. Then the statement $\partial_\mu j^\mu=0$ becomes
$$
\frac{\partial \rho}{\partial t}=-\vec\nabla\cdot\vec{j}
$$
This says that the divergence of the current $\vec{j}$ is equal and opposite to the change in the density. Put another way, if you consider an infinitesimal volume, the current flux out of the boundary of the volume exactly equals the decrease in the density within the volume. This is usually what we mean when we say something (say a fluid) is conserved: If the fluid is flowing out of an area, the density (concentration of the fluid) decreases proportionately. 
Note that if you assume that all currents go to zero at infinity, you can also make a global claim that you might recognize more as a conservation law:
$$
\frac{\partial}{\partial t}\int \rho = -\int\vec\nabla\cdot\vec j= 0
$$
In other words, $\int\rho$ is globally conserved. The first statement is usually referred to as local conservation, since it specifies that any decrease in $\rho$ has to be caused by a current flux, and doesn't allow $\rho$ to spontaneously decrease in one region and increase in another region far away.
A: The charge is $Q=\int d\mathbf{x} j^0$ so $\dot{Q} = \int d\mathbf{x}\partial_0 j^0=-\int d\mathbf{x}\partial_i j^i$. Integrating by parts, $\dot{Q}$ is a boundary term that vanishes if $\mathbf{j}$ vanishes fast as $r\to\infty$. Then $Q$ is conserved.
