  We study the correlations between avalanches in the depinning dynamics of
elastic interfaces driven on a random substrate. In the mean field theory (the
Brownian force model), it is known that the avalanches are uncorrelated. Here
we obtain a simple field theory which describes the first deviations from this
uncorrelated behavior in a $\epsilon=d_c-d$ expansion below the upper critical
dimension $d_c$ of the model. We apply it to calculate the correlations between
(i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or
more generally, in two avalanches separated by a uniform displacement $W$ of
the interface. For (i) we obtain the correlations of the total sizes, of the
local sizes and of the total sizes with given seeds (starting points). For (ii)
we obtain the correlations of the velocities, of the durations, and of the
avalanche shapes. In general we find that the avalanches are {\it
anti-correlated}, the occurence of a larger avalanche making more likely the
occurence of a smaller one, and vice-versa. Examining the universality of our
results leads us to conjecture several new exact scaling relations for the
critical exponents that characterize the different distributions of
correlations. The avalanche size predictions are confronted to numerical
simulations for a $d=1$ interface with short range elasticity. They are also
compared to our recent related work on static avalanches (shocks). Finally we
show that the naive extrapolation of our result into the thermally activated
creep regime at finite temperature, predicts strong positive correlations
between the forward motion events, as recently observed in numerical
simulations.
