Summary: We look for solutions \( u\left( x,t\right) \) of the one-dimensional heat equation \( u_t = u_{xx} \) which are space-time periodic, i.e. they satisfy the property \( u\left( x+a,t+b\right) = u\left( x,t\right)\) for all \(\left( x,t\right) \in\left( -\infty,\infty\right) \times\left( -\infty,\infty\right), \) and derive their Fourier series expansions. Here \( a\geq0, b\geq 0 \) are two constants with \( a^2+b^2>0.\) For general equation of the form \( u_t = u_{xx}+Au_x+Bu, \) where \(A, B \) are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when \( B>0\) and is given by a linear combination of \( \cos\left( \sqrt{B}\left( x+At\right) \right)\) and \(\sin\left( \sqrt{B}\left( x+At\right) \right)\).