Mathematical Logic:
Logic done with
mathematical methods. Aristotle
discovered or invented logic as a science,
and indeed also introduced variables in it, in a
mathematical way, but so far as mathematics and its methods and
notations are concerned, this is were logic
remained the next 2000 years or so.
Leibniz was the first  probably inspired by Pascal and Arnauld  to
think of logic as a mathematical subject, but he did not publish his work.
Mathematical logic was created in the 19th Century, by George Boole 
see his The Mathematical Analysis of the Laws of Thought  and then
further developed by Jevons, Peirce, and Frege. Frege also seems to have been
the first logicist i.e. someone who held that mathematics in fact can be
derived from and founded on logic.
Frege's work, and also Peirce's, was hardly reviewed or known during their
lifetimes, and the first widely known work in mathematical logic was the
Principia Mathematica, by Whitehead and Russell, that attempted  in three
large volumes, mostly filled with mathematical notation, to provide a foundation
for both mathematics and logic, and to derive mathematics from logic.
The Principia Mathematica was a failure in that Whitehead and Russell
found that they could not do without axioms  such as the axioms of infinity and
choice  that were not logical in any clear intuitive sense and provide a
foundation for mathematics, and also required assumptions  as are involved in
the theory of types  that seem neither necessary nor mathematical, but
were introduced by Whitehead and Russell to prevent the arisal of paradoxes,
such as Russell's Paradox, that exploded Frege's system of logic and
mathematics.
Since then mathematical logic has been rapidly developed, mostly by
mathematicians, and some philosophers also. An important text was Hilbert and
Bernays "Grundlagen der Mathematik", and important theorems of the
twenties and thirties were those of Skolem and Gödel (completeness and
incompleteness theorems) whereas Church, Turing, and Tarski
articulated important theories and concepts (lambdaabstraction,
effective computability, and truth and model theory).
The work of Gödel (incompleteness
theorems) refuted the attempts of Hilbert and Bernays to give mathematics a
prooftheoretical foundation, in which
every true mathematical proposition would get its proof.
The work of Church, Turing and Tarski sparked off a great amount of work,
much of it quite mathematical and beyond nonspecialists, in the second half of
the 20th C, and gave rise to electrical programmable computers (Turing,
Von Neumann),
to programming languages (such as Lisp, founded on lambdaabstraction)
and many subtle results about proofs and models, that refined Gödel's and
Tarski's ideas (see Smullyan).
Another logical subject that was mathematiced in the second half of the 20th
C was modal logic, especially after Kripke showed how this could be done
using model theoretical tools.
Also, having mentioned logicism, three schools of thought arose about the
foundations of mathematics in the 20th Century, namely the logicists 
Russell, Carnap  who believed mathematics could be founded on logic; the
intuitionists  Brouwer, Weyl, Heyting  who denied it could, and who also
denied the logical validity of what where up to
then accepted logical axioms (viz. (~~p > p) and (~p V p)), in the end on the
ground that one could only regarded as proved what one could show effectively
(constructively) how to construe; and the formalists (Curry, Feferman,
Boolos) who considered both mathematics and logic as precise formalized kinds of
reasoning with symbols, only constrained by the need for consistency.
There were several important foundational results in the 20th Century, such
as the proof that the Axiom of Choice is independent of the other axioms
of standard set theory (Gödel, Cohen); the discovery of the nonstandard
reals as an alternative foundation for the calculus (Robinson); the
discovery of tableaux as a system of proofs (Beth, Smullyan); the
refinement of Gödel's ideas (Smullyan, Hodges); the proposal of mereology
as a foundation of mathematics (Lesniewski, Lewis); and the mathematical
analysis of quite a few foundational concepts of logic and mathematics (computability,
recursiveness, consequence, probability, proof,
necessity, possibility, constructiveness, programming
languages, categories).
Even so, some very important questions were not solved, notably the
Continuum Problem that already puzzled Cantor; the problem whether standard
set theory is consistent; and the question what is the best or proper foundation
of mathematics.
As to the last two questions:
There is wide agreement among mathematicians and logicians that standard set
theory is consistent, and there are many proofs, for many systems X, that system X is consistent if
standard set theory is (relative consistency), but there has been no proof that standard set theory is
consistent,
and certainly also no proof that it isn't.
And there is also wide agreement that standard set theory is sufficient to
derive most of classical mathematics, and
that set theory and its notation (of which
there are several accepted systems) is very useful as a lingua franca for
mathematics.
But one problem of standard set theory is the proliferation of an
infinity of infinities that it implies (by the
power set axiom and the axiom of infinity), that
is both difficult to make sense of intuitively and to combine with
Ockham's Razor, and apart from that
there have been various other proposals of foundations for mathematics, such as
Curry's formalism (and Schönfinkel and his Combinatory Logic) and
Category Theory, first proposed by McLane.
It would seem that in the 21st Century, the subject that was once
mathematical logic, and done on paper by a few very abstractedly minded pure
mathematicians, is now mostly practised by and taught to computer scientists,
next to the fact that the lingua franca of mathematics is still set
theory, and effectively taught to all who learn mathematics at university level.
In any case, mathematical logic, that may be said to have started with Boole,
has been shown to have amazing practical and theoretical consequences in the
20th Century, and has had huge effects on human civilization and its chances of
survival (computing, internet).
