Many studies have shown that verb inflections are difficult to produce for agrammatic aphasic speakers: they are frequently omitted and substituted. The present article gives an overview of our search to understanding why this is the case. The hypothesis is that grammatical morphology referring to the past is selectively impaired in agrammatic aphasia. That is, verb inflections for past tense and perfect aspect are hard to produce. Furthermore, verb clusters that refer to the past will be affected as well, even if the auxiliary is in present tense, as in he has been writing a letter. It will be argued that all these verb forms referring to the past require discourse linking [Zagona, K. (2003). Tense and anaphora: Is there a tense-specific theory of coreference. In A. Barrs (Ed.), "Anaphora: A reference guide" (pp. 140-171). Oxford: Blackwell Publishing] and discourse linking is affected in agrammatic aphasia [Avrutin, S. (2006). Weak syntax. In K. Amunts, & Y. Grodzinsky (Eds.), "Broca's region" (pp. 49-62). New York: Oxford Press]. This hypothesis has been coined the PAst DIscourse LInking Hypothesis (PADILIH) [Bastiaanse, R., Bamyaci, E., Hsu, C.-J., Lee, J., Yarbay Duman, T., & Thompson, C.K. (2011). Time reference in agrammatic aphasia: A cross-linguistic study. "Journal of Neurolinguistics, 24", 652-673]. The PADILIH has been tested in several languages and populations that have hardly been studied before in aphasiology: languages such as Turkish, Swahili and Indonesian were included, as well as monolingual and bilingual populations. In all these populations, the same test has been used: the "Test for Assessing Reference of Time" [Bastiaanse, R., Jonkers, R., & Thompson, C.K. (2008). "Test for assessing reference of time" (TART). Groningen: University of Groningen] to enable reliable comparisons between the languages. The results show that the PADILIH predicts the performance of agrammatic speakers very well: discourse-linked grammatical morphemes expressing time reference to the past are hard to produce for agrammatic speakers, whereas non-discourse-linked verb inflections (for present and future) are relatively spared. In languages that use aspectual adverbs (free-standing and optional time reference markers), such as Chinese and Indonesian, time reference to all time frames is impaired, since all aspectual adverbs, regardless of the time frame they refer to, require discourse linking. Remarkably, the problems are not restricted to grammatical morphemes: the production of temporal lexical adverbs is impaired as well. (Contains 12 figures and 3 notes.)