Important constraints on the properties of the anomalous X-ray pulsars (AXP
s) and soft gamma-ray repeaters (SGRs) can be provided by their association
s with supernova remnants (SNRs). We have made a radio search for SNRs towa
rd the AXPs RX J170849-400910 and 4U 0142+61. We find that the former lies
near a possible new SNR with which it is unlikely to be physically associat
ed, but we see no SNR in the vicinity of the latter. We review all claimed
pairings between AXPs and SNRs; the three convincing associations imply tha
t AXPs are young (<10 000 yr) neutron stars with low projected space veloci
ties (<500 km s(-1)). Contrary to previous claims, we find no evidence that
the density of the ambient medium around AXPs is higher than that in the v
icinity of radio pulsars. Furthermore, the nondetection of radio emission f
rom AXPs does not imply that these sources are radio-silent. We also review
claimed associations between SGRs and SNRs. We find none of these associat
ions to be convincing, consistent with a scenario in which SGRs and AXPs ar
e both populations of high-field neutron stars ("magnetars") but the SGRs a
re an older or longer-lived group of objects than are the AXPs. If the SGR-
SNR associations are shown to be valid, then SGRs must be high-velocity obj
ects and most likely represent a different class of source to the AXPs.