This paper introduces symmetric, informationally complete positive operator-valued measures (SIC-POVMs), which are sets of $ d^2 $ rank-one operators in $ d $-dimensional Hilbert space with equal pairwise inner products. These are equivalent to $ d^2 $ equiangular lines in $ \mathbb{C}^d $ and are relevant for quantum state tomography, quantum cryptography, and foundational quantum mechanics. The authors construct SIC-POVMs in dimensions 2, 3, and 4, and conjecture that group-covariant SIC-POVMs exist in all finite dimensions. They provide numerical results up to dimension 45, supporting this conjecture. SIC-POVMs are also related to spherical t-designs. A spherical t-design is a set of vectors such that the average of any t-th order polynomial over the set equals the average over all normalized vectors. The paper shows that SIC-POVMs are 2-designs, and that every 2-design with $ d^2 $ elements is a SIC-POVM. The authors also explore group-covariant SIC-POVMs, where the set is invariant under a group action and the group acts transitively on the set. They use the group $ Z_d \times Z_d $ to construct SIC-POVMs in dimensions 2, 3, and 4, and find that there are two, an uncountably infinite number, and 16 such SIC-POVMs, respectively. Numerical methods are used to find SIC-POVMs in higher dimensions, minimizing the second frame potential. The authors find $ Z_d \times Z_d $-covariant solutions for dimensions 5 through 45. They also test other nice error bases and find that some generate SIC-POVMs. The paper concludes that SIC-POVMs likely exist in all finite dimensions, and they are important for quantum information theory, including quantum state tomography, quantum cryptography, and error-correcting codes. They are also connected to mathematical problems such as spherical codes and mutually unbiased bases.This paper introduces symmetric, informationally complete positive operator-valued measures (SIC-POVMs), which are sets of $ d^2 $ rank-one operators in $ d $-dimensional Hilbert space with equal pairwise inner products. These are equivalent to $ d^2 $ equiangular lines in $ \mathbb{C}^d $ and are relevant for quantum state tomography, quantum cryptography, and foundational quantum mechanics. The authors construct SIC-POVMs in dimensions 2, 3, and 4, and conjecture that group-covariant SIC-POVMs exist in all finite dimensions. They provide numerical results up to dimension 45, supporting this conjecture. SIC-POVMs are also related to spherical t-designs. A spherical t-design is a set of vectors such that the average of any t-th order polynomial over the set equals the average over all normalized vectors. The paper shows that SIC-POVMs are 2-designs, and that every 2-design with $ d^2 $ elements is a SIC-POVM. The authors also explore group-covariant SIC-POVMs, where the set is invariant under a group action and the group acts transitively on the set. They use the group $ Z_d \times Z_d $ to construct SIC-POVMs in dimensions 2, 3, and 4, and find that there are two, an uncountably infinite number, and 16 such SIC-POVMs, respectively. Numerical methods are used to find SIC-POVMs in higher dimensions, minimizing the second frame potential. The authors find $ Z_d \times Z_d $-covariant solutions for dimensions 5 through 45. They also test other nice error bases and find that some generate SIC-POVMs. The paper concludes that SIC-POVMs likely exist in all finite dimensions, and they are important for quantum information theory, including quantum state tomography, quantum cryptography, and error-correcting codes. They are also connected to mathematical problems such as spherical codes and mutually unbiased bases.