We present TRIÈST, a suite of one-pass streaming algorithms to compute unbiased, low-variance, high-quality approximations of the global and local (i.e., incident to each vertex) number of triangles in a fully-dynamic graph represented as an adversarial stream of edge insertions and deletions. Our algorithms use reservoir sampling and its variants to exploit the user-specified memory space at all times. This is in contrast with previous approaches which use hard-to-choose parameters (e.g., a fixed sampling probability) and offer no guarantees on the amount of memory they will use. We show a full analysis of the variance of the estimations and novel concentration bounds for these quantities. Our experimental results on very large graphs show that TRIÈST outperforms state-of-the-art approaches in accuracy and exhibits a small update time.