  We propose a new simplified definition of extended affine Lie algebras (EALAs
for short), and also discuss a general version of extended affine Lie algebras,
called locally extended affine Lie algebras (LEALAs for short). We prove a
conjecture by V. Kac for LEALAs. It turns out that the root system of a LEALA
becomes a locally finite version of extended affine root systems. Finally,
several examples of new EALAs and LEALAs are introduced, and we classify LEALAs
of nullity 0 with connection to locally finite split simple Lie algebras.
