The talk will be focused on central-upwind schemes, which are simple, efficient, highly accurate and robust Godunov-type finite-volume methods for hyperbolic systems of conservation and balance laws; see, e.g., the review papers [1,2]. I will first briefly go over the main three steps in the derivation of central-upwind schemes. First, we assume that the computed solution is realized in terms of its cell averages, which are used to construct a global in space piecewise polynomial interpolant. We then evolve the computed solution according to the integral form of the studied hyperbolic system. The evolution is performed using a nonsymmetric set of control volumes, whose size is proportional to the local speeds of propagation: this allow one to avoid solving any (generalized) Riemann problems. Once the solution is evolved, it must be projected back onto the original grid as otherwise the number of evolved cell averages would double every time step and the scheme would become impractical. The projection should be carried out in a very careful manner as the projection step may bring an excessive amount of numerical dissipation into the resulting scheme as was the case in previous versions of the central-upwind schemes. In order to more accurately project the solution, we have recently introduced a new way of making the projection. A major novelty of the new approach is that we use a subcell resolution and reconstruct the solution at each cell interface using two linear pieces. This allows us to perform the projection in the way, which would be extremely accurate in the vicinities of linearly degenerate contact waves. This leads to the new second-order semi-discrete low-dissipation central-upwind schemes [3,4], which clearly outperform their existing counterparts as confirmed by a number of numerical experiments. REFERENCES [1] A. Kurganov. Central schemes: A powerful black-box solver for nonlinear hyperbolic PDEs. Handbook of numerical methods for hyperbolic problems, Handb. Numer. Anal., Vol. 17, pp. 525–548, Elsevier/North-Holland, Amsterdam, 2016. [2] A. Kurganov. Finite-volume schemes for shallow-water equations. Acta Numer., Vol. 27, pp. 289–351, 2018. [3] A. Kurganov, R. Xin. New low-dissipation central-upwind schemes. J. Sci. Comput., Vol. 96, Paper No. 56, 2023. [4] S. Chu, A. Kurganov, R. Xin. New low-dissipation central-upwind schemes. Part II. Submitted, preprint available at http://arxiv.org/abs/2405.07620.