I am certainly sure that any one who has read Gil Kalai's witty community wiki has benefited a lot.
Here I follow a similar track in asking this question.
So let's compose a list of fundamental theorems in mathematics which may not even have the tag "fundamental" but have serious wight in the respective branch of math.


I will start with the elementary and very popular ones.(Please add a description if the theorem is fundamental but still not so well-known)


Thanks for all your effort.


1. FTA: The Fundamental Theorem of Arithmetic (or Unique-Prime-Factorization Theorem):
 ->Any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers.
2. FTA: The Fundamental theorem of Algebra:
 -> The field of complex numbers is algebraically closed
3. FTC: The fundamental theorem of calculus:
 -> Has two parts and specifies the relationship between the two central operations of calculus: differentiation and integration.
4. FTLP: The fundamental theorem of linear programming:
 -> In a weak formulation, states that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners.



Fundamental theorems 
In his book *Topics in Geometric Group Theory*, Pierre de la Harpe calls the following result the **Fundamental Observation of Geometric Group Theory** (though he also calls it a theorem!). It is also often called the Svarc--Milnor Lemma. Roughly speaking, it asserts that the coarse geometry of a group is captured by any suitably nice action of that group by isometries on a metric space.


**Theorem.** Let $X$ be a metric space that is geodesic and proper, let $\Gamma$ be a group and let $\Gamma$ act properly discontinuously and cocompactly by isometries on $X$. Then $\Gamma$ is finitely generated, and furthermore for any $x\_0\in X$ the map $\Gamma\to X$ given by


$\gamma\mapsto\gamma x\_0$


is a quasi-isometry.


**Remarks.**


1. $\Gamma$ is endowed with the word metric (with respect to some choice of finite generating set).
2. A map of metric space $f:Y\to X$ is a *quasi-isometric embedding* if there are constants $\lambda\geq 1$, $\mu\geq 0$ such that


$\lambda d\_Y(y\_1,y\_2)+\mu\geq d\_X(f(y\_1),f(y\_2))\geq \frac{1}{\lambda} d\_Y(y\_1,y\_2)-\mu$


for all $y\_1,y\_2\in Y$. It is a *quasi-isometry* if, furthermore, for every $x\in X$ there is $y\in Y$ such that $d(x,f(y))\leq \mu$.


