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Let $Y$ be an affine algebraic variety, and let $X$ be a smooth resolution of singularities of the variety $Y$. Singularities of $Y$ are said to be symplectic if $X/Y$ carries a closed holomorphic 2-form which is non-degenerate outside of the exceptional set. It was discovered recently by A. Beauville, Y. Namikawa, J. Wierzba and others that symplectic singularities possess a rich internal structure -- it is possible to claim and prove much stronger results than one would expect from the viewpoint of usual algebraic geometry. We will give an overview of these recent advances and outline directions of further research. We will pay particular attention to the especially interesting case when the symplectic form on $X$ is non-degenerate everywhere, and even stronger, to the case when the variety $Y$ is the quotient $Y=V/G$ of a symplectic vector space $V$ by an action of a finite group $G$.