<p>This survey explores the spaces <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(A^{-\infty }(\varOmega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mrow> <mo>-</mo> <mi>∞</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and their duals, focusing on a broad class of domains <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\varOmega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Ω</mi> </math></EquationSource> </InlineEquation>. These include not only the unit disc <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb {D}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> and the unit ball <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathbb {B}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">B</mi> </math></EquationSource> </InlineEquation>, but also bounded convex domains in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathbb {C}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and, more generally, linearly convex domains in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\mathbb {C}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. Special attention is given to applications of duality theory, particularly to the representation of functions in these spaces by exponential or rational functions, and to their connections with convolution equations.</p>