  We present a general central limit theorem with simple, easy-to-check
covariance-based sufficient conditions for triangular arrays of random vectors
when all variables could be interdependent. The result is constructed from
Stein's method, but the conditions are distinct from related work. We show that
these covariance conditions nest standard assumptions studied in the literature
such as $M$-dependence, mixing random fields, non-mixing autoregressive
processes, and dependency graphs, which themselves need not imply each other.
This permits researchers to work with high-level but intuitive conditions based
on overall correlation instead of more complicated and restrictive conditions
such as strong mixing in random fields that may not have any obvious
micro-foundation. As examples of the implications, we show how the theorem
implies asymptotic normality in estimating: treatment effects with spillovers
in more settings than previously admitted, covariance matrices, processes with
global dependencies such as epidemic spread and information diffusion, and
spatial process with Mat\'{e}rn dependencies.
