  The penetration of magnetic flux into a thin film of YBa$_2$Cu$_3$O$_{7-x}$
is studied when the external field is ramped slowly. In this case the flux
penetrates in bursts or avalanches. The size of these avalanches is distributed
according to a power law with an exponent of $\tau$ = 1.29(2). The additional
observation of finite-size scaling of the avalanche distributions, with an
avalanche dimension D = 1.89(3), gives strong indications towards
self-organized criticality in this system. Furthermore we determine exponents
governing the roughening dynamics of the flux surface using some universal
scaling relations. These exponents are compared to those obtained from a
standard roughening analysis.
