We will work over $\mathbb{C}$.
Remember that a finite subgroup $G$ of $\operatorname{GL}_n(\mathbb{C})$ is called a complex reflection group if it is generated by complex reflections $r$, which are linear automorphisms of finite order such that $\operatorname{rank} I - r= 1$; as usual, using Weyl's trick, we can suppose that $G$ preservers a positive definite Hermitian form on $\mathbb{C}^n$. Those groups were classified in the classical work of Shephard and Todd, and have a myriad of applications in many branches of mathematics, such as representation theory of linear algebraic groups, knot theory, Hecke algebras, combinatorics, invariant theory, etc. See, for instance, the books Unitary reflection groups by Lehrer, or Introduction to complex reflection groups and their braid groups by Broué.
Symplectic reflection groups can be seen as an analogue of complex reflection groups in a symplectic vector space $(V, \omega)$, where $\omega$ is a non-degenerate skew-symmetric form on $V$. If $r \in \operatorname{Sp}(V)$ is an element of finite order, it cannot be a complex reflection, since the determinant of a complex reflection is a root of unity $\lambda \neq 1$, while elements of $\operatorname{Sp}(V)$ have determinant 1. An element $r$ of $\operatorname{Sp}(V)$ is called a symplectic reflection if the next best situation happens: $\operatorname{rank} I - r =2$. And a finite subgroup $G$ of $\operatorname{Sp}(V)$ is called a symplectic reflection group if it is generated by symplectic reflections. Those were classified by Cohen in Finite quaternionic reflection groups.
Symplectic reflection groups are relatively new in comparison with the complex reflection groups, but in recent times, in particular due to their relevance in the study of symplectic singularities and the enormous influence of the theory of symplectic reflection algebras of Etingof and Ginzburg (of which the rational Cherednik algebras are a special case), they received a lot of attention.
I'm looking for surveys or expositions which are focused on the properties of symplectic reflection groups themselves, and not so much about their aplications. In particular, I would like to know if there is a place other then Cohen's paper where I can study their classification.
There is one kind of application of symplectic reflection groups which I am interested though: invariant theory. Now, the role of complex reflection groups in invariant theory is extremely important and well known: it is the content of the Chevalley-Shephard-Todd Theorem (when the base field is the complex numbers). I'm wondering if symplectic reflection groups have known nice results about their invariant theory. If I remember correctly, Kac and Watanabe showed that the invariant subalgebra of a polynomial algebra can be a complete intersection only if the group is generated by bireflections: elements $g$ of finite order such that $\operatorname{rank} I-g=2$, but I am not completely sure about this.
