Comparing spoken segments is a central operation to speech processing. Traditional approaches in this area have favored frame-level dynamic programming algorithms, such as dynamic time warping, because they require no supervision, but they are limited in performance and efficiency. As an alternative, acoustic word embeddings -- fixed-dimensional vector representations of variable-length spoken word segments -- have begun to be considered for such tasks as well. However, the current space of such discriminative embedding models, training approaches, and their application to real-world downstream tasks is limited. We start by considering ``single-view" training losses where the goal is to learn an acoustic word embedding model that separates same-word and different-word spoken segment pairs. Then, we consider ``multi-view" contrastive losses. In this setting, acoustic word embeddings are learned jointly with embeddings of character sequences to generate acoustically grounded embeddings of written words, or acoustically grounded word embeddings. In this thesis, we contribute new discriminative acoustic word embedding (AWE) and acoustically grounded word embedding (AGWE) approaches based on recurrent neural networks (RNNs). We improve model training in terms of both efficiency and performance. We take these developments beyond English to several low-resource languages and show that multilingual training improves performance when labeled data is limited. We apply our embedding models, both monolingual and multilingual, to the downstream tasks of query-by-example speech search and automatic speech recognition. Finally, we show how our embedding approaches compare with and complement more recent self-supervised speech models.