  Quenched disorder - in the sense of the Harris criterion - is generally a
relevant perturbation at an absorbing state phase transition point. Here using
a strong disorder renormalization group framework and effective numerical
methods we study the properties of random fixed points for systems in the
directed percolation universality class. For strong enough disorder the
critical behavior is found to be controlled by a strong disorder fixed point,
which is isomorph with the fixed point of random quantum Ising systems. In this
fixed point dynamical correlations are logarithmically slow and the static
critical exponents are conjecturedly exact for one-dimensional systems. The
renormalization group scenario is confronted with numerical results on the
random contact process in one and two dimensions and satisfactory agreement is
found. For weaker disorder the numerical results indicate static critical
exponents which vary with the strength of disorder, whereas the dynamical
correlations are compatible with two possible scenarios. Either they follow a
power-law decay with a varying dynamical exponent, like in random quantum
systems, or the dynamical correlations are logarithmically slow even for weak
disorder. For models in the parity conserving universality class there is no
strong disorder fixed point according to our renormalization group analysis.
