  In this review article we discuss a number of recent results concerning wild
weak solutions of the incompressible Euler and Navier-Stokes equations. These
results build on the groundbreaking works of De Lellis and Sz\'ekelyhidi Jr.,
who extended Nash's fundamental ideas on $C^1$ flexible isometric embeddings,
into the realm of fluid dynamics. These techniques, which go under the umbrella
name convex integration, have fundamental analogies the phenomenological
theories of hydrodynamic turbulence. Mathematical problems arising in
turbulence (such as the Onsager conjecture) have not only sparked new interest
in convex integration, but certain experimentally observed features of
turbulent flows (such as intermittency) have also informed new convex
integration constructions.
  First, we give an elementary construction of nonconservative $C^{0+}_{x,t}$
weak solutions of the Euler equations, first proven by De Lellis-Sz\'ekelyhidi
Jr.. Second, we present Isett's recent resolution of the flexible side of the
Onsager conjecture. Here, we in fact follow the joint work of De
Lellis-Sz\'ekelyhidi Jr. and the authors of this paper, in which weak solutions
of the Euler equations in the regularity class $C^{\frac 13-}_{x,t}$ are
constructed, attaining any energy profile. Third, we give a concise proof of
the authors' recent result, which proves the existence of infinitely many weak
solutions of the Navier-Stokes in the regularity class $C^0_t L^{2+}_x \cap
C^0_t W^{1,1+}_x$. We conclude the article by mentioning a number of open
problems at the intersection of convex integration and hydrodynamic turbulence.
