Tensor algebra provides a mathematical language prevalent in numerous domains of computing, including computational chemistry, machine learning, and quantum information. The Cyclops library provides high-performance algorithms for fundamental operations (summation, contraction, factorization, slicing, reshaping, etc.) on dense or sparse tensors. The library uses distributed storage via MPI and executes each tensor operation bulk synchronously. Algebraic tensor operations are specified via a high-level Einstein summation syntax, accessible as a standalone library in C++ or Python. Cyclops uses high-performance communication-avoiding algorithms for (sparse) matrix multiplication, optimized transposition/redistribution routines, and runtime mapping/algorithm selection based on trainable performance models. The library has been used to break computational frontiers in electronic structure calculations and is part of QChem, CC4S, and PySCF. It has also been used for large-scale quantum circuit simulation as well as other applications such as graph analysis.