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We study the capacity of discrete memoryless channels with synchronization errors and additive noise. We first show that with very large alphabets, their capacity can be achieved by independent and identically distributed input sources, and establish proven tight lower and upper capacity bounds. We also derive tight numerical capacity bounds for channels where the synchronization between the input and output is partly preserved, for instance using incorruptible synchronization markers. Such channels include channels with duplication errors, channels that only insert or delete zeros, and channels with bitshift errors studied in magnetic recording. Channels with small alphabets and corrupted by synchronization errors have an infinite memory. Revisiting the theoretical work of Dobrushin and adapting techniques used to compute capacity bounds for finite-state source/channel models, we compute improved numerical capacity lower bounds for discrete memoryless channels with small alphabets, synchronization errors, and memoryless noise. An interesting and some- what surprising result is that as long as the input sequences are not completely deleted, the capacity of channels corrupted by discrete timing errors is always nonzero even if all the symbols are corrupted.