In a recent paper by \textit{V. D. Milman} and the author [Isr. J. Math. 54, 139-158 (1986; Zbl 0611.46022)] the notion of weak cotype 2 and weak type 2 Banach spaces were introduced. In the present paper the author considers the class of Banach spaces which are both of weak type 2 and weak cotype 2. These spaces are referred to as weak Hilbert spaces since Hilbert spaces themselves are characterized as both of type 2 and cotype 2. In the paper the author gives several equivalent characterizations and a number of properties of these spaces. For example, it is shown that weak Hilbert spaces are reflexive and possess the approximation property. The paper also suggests numerous questions which deserve further investigation.