Duality-Symmetric Approach to General Relativity and Supergravity

We review the application of a duality-symmetric approach to gravity and supergravity with emphasizing benefits and disadvantages of the formulation. Contents of these notes includes: 1) Introduction with putting the accent on the role of dual gravity within $M$-theory; 2) Dualization of gravity with a cosmological constant in $\mathrm D=3$; 3) On-shell description of dual gravity in $\mathrm D>3$; 4) Construction of the duality-symmetric action for General Relativity with/without matter fields; 5) On-shell description of dual gravity in linearized approximation; 6) Brief summary of the paper.