We argue that rich data gathered in experimental primatology in the last 40 years can benefit from analytical methods used in contemporary linguistics. Focusing on the syntactic and especially semantic side, we suggest that these (iii) how are the meanings of individual calls combined? (iv) how do calls or call sequences compete with each other when several are appropriate in a given situation? (v) how did the form and meaning of calls evolve? We address these questions in five case studies pertaining to cercopithecines (Putty-nosed monkeys, Blue monkeys, and Campbell's monkeys), colobinae (Guereza monkeys and King Colobus monkeys), and New World monkeys (Titi monkeys). The morphology mostly involves simple calls, but in at least one case (Campbell's -oo) we find a root-suffix structure, possibly with a compositional semantics. The syntax is in all clear cases simple and finite-state. With respect to meaning, nearly all cases of call concatenation can be analyzed as conjunction. But a key question concerns the division of labor between semantics, pragmatics and the environmental context ('world' knowledge and context change). An apparent case of dialectal variation in the semantics (Campbell's krak) can arguably be analyzed away if one posits sufficiently powerful mechanisms of competition among calls, akin to scalar implicatures. An apparent case of non-compositionality (Putty-nosed pyow-hack sequences) can be analyzed away if one further posits a pragmatic principle of 'urgency', whereby threat-related calls must come early in sequences (another potential case of non-compositionality -Colobus snort-roar sequences -might justify assigning non-compositional meanings to complex calls, but results are tentative). Finally, rich Titi sequences in which two calls are re-arranged in complex ways so as to reflect information about both predator identity and location are argued not to involve a complex syntax/semantics interface, but rather a finegrained interaction between simple call meanings and the environmental context. With respect to call evolution, we suggest that the remarkable preservation of call form and function over millions of years should make it possible to lay the groundwork for an evolutionary monkey linguistics, which we illustrate with cercopithecine booms, and with a comparative analysis of Blue monkey and Putty-nosed monkey repertoires. Throughout, we aim to compare possible theories rather than to fully adjudicate between them, and our claims are correspondingly modest. But we hope that our methods could lay the groundwork for a formal monkey linguistics combining data from primatology with formal techniques from linguistics (from which it does not follow that the calls under study share non-trivial properties, let alone an evolutionary history, with human language).