Title: On the Linguistic Capacity of Real-time Counter Automata. Abstract: While counter machines have received little attention in theoretical computer science since the 1960s, they have recently achieved a newfound relevance to the field of natural language processing (NLP). Recent work has suggested that some strong-performing recurrent neural networks utilize their memory as counters. Thus, one potential way to understand the sucess of these networks is to revisit the theory of counter computation. Therefore, we choose to study the abilities of real-time counter machines as formal grammars. We first show that several variants of the counter machine converge to express the same class of formal languages. We also prove that counter languages are closed under complement, union, intersection, and many other common set operations. Next, we show that counter machines cannot evaluate boolean expressions, even though they can weakly validate their syntax. This has implications for the interpretability and evaluation of neural network systems: successfully matching syntactic patterns does not guarantee that a counter-like model accurately represents underlying semantic structures. Finally, we consider the question of whether counter languages are semilinear. This work makes general contributions to the theory of formal languages that are of particular interest for the interpretability of recurrent neural networks.