The brown rat, Rattus norvegicus, is both a notorious pest and a frequently used model in biomedical research. By analysing genome sequences of 12 wild-caught brown rats from their ancestral range in NE China, along with the sequence of a black rat, R. rattus, we investigate the selective and demographic forces shaping variation in the genome. We estimate that the recent effective population size (N_e) of this species = 1.24 x 10^5, based on silent site diversity. We compare patterns of diversity in these genomes with patterns in multiple genome sequences of the house mouse Mus musculus castaneus), which has a much larger N_e. This reveals an important role for variation in the strength of genetic drift in mammalian genome evolution. By a Pairwise Sequentially Markovian Coalescent (PSMC) analysis of demographic history, we infer that there has been a recent population size bottleneck in wild rats, which we date to approximately 20,000 years ago. Consistent with this, wild rat populations have experienced an increased flux of mildly deleterious mutations, which segregate at higher frequencies in protein-coding genes and conserved noncoding elements (CNEs). This leads to negative estimates of the rate of adaptive evolution (alpha) in proteins and CNEs, a result which we discuss in relation to the strongly positive estimates observed in wild house mice. As a consequence of the population bottleneck, wild rats also show a markedly slower decay of linkage disequilibrium with physical distance than wild house mice.