This chapter reviews the past research toward identifying the brain circuit and its computation underlying the associative memory in eyeblink classical conditioning. In the standard delay eyeblink conditioning paradigm, the conditioned stimulus (CS) and eyeblink-eliciting unconditioned stimulus (US) converge in the cerebellar cortex and interpositus nucleus (IPN) through the pontine nuclei and inferior olivary nucleus. Repeated pairings of CS and US modify synaptic weights in the cerebellar cortex and IPN, enabling IPN neurons to activate the red nucleus and generate the conditioned response (CR). In a variant of the standard paradigm, trace eyeblink conditioning, the CS and US are separated by a brief stimulus-free trace interval. Acquisition in trace eyeblink conditioning depends on several forebrain regions, including the hippocampus and medial prefrontal cortex as well as the cerebellar-brainstem circuit. Details of computations taking place in these regions remain unclear; however, recent evidence supports a view that the forebrain encodes a temporal sequence of the CS, trace interval, and US in a specific environmental context and signals the cerebellar-brainstem circuit to execute the CR when the US is likely to occur. Together, delay eyeblink conditioning represents one of the most successful cases of understanding the neural substrates of long-term memory in mammals, while trace eyeblink conditioning demonstrates its utility for uncovering detailed computations in the whole brain network underlying long-term memory.