In this paper we study composition operators on spaces of entire functions. We determine which entire functions induce bounded composition operators on the Paley-Wiener space, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript pi Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">L_\pi ^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E squared left-parenthesis gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>γ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E^2(\gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces. In addition, we characterize compact composition operators on these spaces. We also study the cyclic properties of composition operators acting on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript pi Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">L_\pi ^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.