In this thesis, we develop high performance algorithms for certain computations involving dense tensors and sparse matrices. We address kernel operations that are useful for machine learning tasks, such as inference with deep neural networks (DNNs). We develop data structures and techniques to reduce memory use, to improve data locality and hence to improve cache reuse of the kernel operations. We design both sequential and shared-memory parallel algorithms. In the first part of the thesis we focus on dense tensors kernels. Tensor kernels include the tensor--vector multiplication (TVM), tensor--matrix multiplication (TMM), and tensor--tensor multiplication (TTM). Among these, TVM is the most bandwidth-bound and constitutes a building block for many algorithms. We focus on this operation and develop a data structure and sequential and parallel algorithms for it. We propose a novel data structure which stores the tensor as blocks, which are ordered using the space-filling curve known as the Morton curve (or Z-curve). The key idea consists of dividing the tensor into blocks small enough to fit cache, and storing them according to the Morton order, while keeping a simple, multi-dimensional order on the individual elements within them. Thus, high performance BLAS routines can be used as microkernels for each block. We evaluate our techniques on a set of experiments. The results not only demonstrate superior performance of the proposed approach over the state-of-the-art variants by up to 18%, but also show that the proposed approach induces 71% less sample standard deviation for the TVM across the d possible modes. Finally, we show that our data structure naturally expands to other tensor kernels by demonstrating that it yields up to 38% higher performance for the higher-order power method. Finally, we investigate shared-memory parallel TVM algorithms which use the proposed data structure. Several alternative parallel algorithms were characterized theoretically and implemented using OpenMP to compare them experimentally. Our results on up to 8 socket systems show near peak performance for the proposed algorithm for 2, 3, 4, and 5-dimensional tensors. In the second part of the thesis, we explore the sparse computations in neural networks focusing on the high-performance sparse deep inference problem. The sparse DNN inference is the task of using sparse DNN networks to classify a batch of data elements forming, in our case, a sparse feature matrix. The performance of sparse inference hinges on efficient parallelization of the sparse matrix--sparse matrix multiplication (SpGEMM) repeated for each layer in the inference function. We first characterize efficient sequential SpGEMM algorithms for our use case. We then introduce the model-parallel inference, which uses a two-dimensional partitioning of the weight matrices obtained using the hypergraph partitioning software. The model-parallel variant uses barriers to synchronize at layers. Finally, we introduce tiling model-parallel and tiling hybrid algorithms, which increase cache reuse between the layers, and use a weak synchronization module to hide load imbalance and synchronization costs. We evaluate our techniques on the large network data from the IEEE HPEC 2019 Graph Challenge on shared-memory systems and report up to 2x times speed-up versus the baseline.