  Heavy-ion collisions at relativistic energies probe matter at extreme
conditions of temperatures and energy densities. The study of event-by-event
fluctuations of experimental observables is crucial to probe the QCD phase
transition, locate the critical point, and learn about the associated critical
phenomena. At the critical point, all thermodynamic quantities behave
anomalously. Fluctuation measurements provide access to thermodynamic response
functions. We discuss the methods for obtaining the isothermal compressibility
using particle multiplicity fluctuation, and specific heat using fluctuations
in mean transverse momentum, temperature, and energy. Lattice QCD calculations
have predicted non-monotonic behavior in the higher-order cumulants of
conserved quantities at the critical point. Fluctuations in the multiplicity of
charged to neutral particles have been measured to understand the formation of
domains of disoriented chiral condensates. We review the recent fluctuation
results as a function of collision centrality and energy from experiments at
SPS, RHIC, and LHC. In addition, we propose to map the temperature fluctuations
in eta-phi plane to probe local fluctuations of temperature and energy density.
