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Interest in Na-S compounds stems from their use in battery materials at 1 atm, as well as the potential for superconductivity under pressure. Evolutionary structure searches coupled with Density Functional Theory calculations were employed to predict stable and low-lying metastable phases of sodium poor and sodium rich sulfides at 1 atm and within 100&#8722;200 GPa. At ambient pressures, four new stable or metastable phases with unbranched sulfur motifs were predicted: Na₂S₃ with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mn>2</mn> <mo>/</mo> <mi>c</mi> </mrow> </semantics> </math> </inline-formula> and <i>Imm</i>2 symmetry, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>-Na₂S₅ and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>-Na₂S₈. Van der Waals interactions were shown to affect the energy ordering of various polymorphs. At high pressure, several novel phases that contained a wide variety of zero-, one-, and two-dimensional sulfur motifs were predicted, and their electronic structures and bonding were analyzed. At 200 GPa, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <mn>4</mn> <mo>/</mo> <mi>m</mi> <mi>m</mi> <mi>m</mi> </mrow> </semantics> </math> </inline-formula>-Na₂S₈ was predicted to become superconducting below 15.5 K, which is close to results previously obtained for the <inline-formula> <math display="inline"> <semantics> <mi>&#946;</mi> </semantics> </math> </inline-formula>-Po phase of elemental sulfur. The structures of the most stable M₃S and M₄S, M = Na, phases differed from those previously reported for compounds with M = H, Li, K.