1. For rising and falling factorials: $x^{\overline{n}}$ and $x_{\underline{n}}$ x^{\underline{n}}$à la Knuth. Much better than the traditional way to write the Pochhammer symbol:$(x)_n := x^{\overline{n}}$. In a book I'm writing, I use the notation$x^{\uparrow n}$and$x^{\downarrow n}$, which I find much less clumsy (consider$(2x+1)^{\overline{6k-2}}$vs$(2x+1)^{\uparrow6k-2}$). Anyway, the utility in either of these notations is seen in the umbral calculus; it makes the connection to "ordinary" calculus much more apparent, such as with $$\Delta x^{\uparrow n} = n x^{\uparrow n-1}\qquad\text{compared to}\qquad D x^n = nx^{n-1}.$$ 2. The simple idea of omitting parentheses for function application:$f\,x$as opposed to$f(x)$. I think this often makes some mathematics look cleaner, especially when the argument isn't especially complex. It also allows for some nice (= convenient) abuse of notation, such as in $$\left[ (-1)^{p - m - n} z \prod_{j = 1}^p \left( z D_z - a_j + 1 \right) - \prod_{j = 1}^q \left( z D_z - b_j \right) \right] G(z) = 0,$$ where$D_z:=d/dz$. Note this equation isn't a product (entirely); upon expansion, we'd have$D_z G(z)$terms. 3. Do fractions count? Imagine having to write $$\sqrt{(x^2 + 2x + 1)\div (5x^3 - 3x^2 + 2x - 7)}$$ instead of $$\sqrt{\frac{x^2 + 2x + 1}{5x^3 - 3x^2 + 2x - 7}}.$$ 4. Big-O notation. Though often abused, this is a much less clumsy way to express boundedness and asymptotics and errors and even lets you begin to do some algebra with them (provided you're careful). I don't think doing such is as obvious when you write it all out manually. 5.$\square(x)$for the square wave,$\triangle(x)$for the triangle wave,$Ш(x)$for the Dirac comb (seriously, see Appel's "Mathematics for Physics and Physicists"). These are more cute than explicitly useful. 6. Notation used with musical isomorphisms as a way to do raising and lowering of indices. We have$X^\sharp$which raises the index (in the context of Einstein summation) and$X^\flat$which lowers the index. Here,$\flat$and$\sharp$are isomorphisms between tangent$TM$and cotangent bundles$T^*M$:$\flat:TM\to T^*M$and$\sharp:T^*M\to TM$. 7. Using$\operatorname{cis}\theta = \cos\theta + \mathrm{i}\sin\theta$(cosine i sine), which is nice for obvious reasons (yes,$\omega = e^{\mathrm{i}\theta}$is nice too) and$\operatorname{cas}\theta = \cos\theta + \sin\theta$(cosine and sine), which is used in e.g., the Hartley transform. 8. Notations for hypergeometric functions $${}_pF_q \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right) = {}_pF_q(\mathbf{a},\mathbf{b};z)$$ and Meijer-$G$functions: $$G_{p,q}^{m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right)=G_{p,q}^{m,n} \!\left( \left. \begin{matrix} \mathbf{a} \\ \mathbf{b} \end{matrix} \; \right| \, z \right)$$ 9. Notation for general continued fractions: $$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$$ The$\mathrm{K}$comes from German's "Kettenbruch", which is "continued fraction." I think that's good for now. There are probably lots more. :) To end, I'll say one notation I do not like: the use of fraktur. Most of the time it just looks ugly and no one can actually write fraktur letters. Post Made Community Wiki by Ben Webster 3 bit more description 1. For rising and falling factorials:$x^{\overline{n}}$and$x_{\underline{n}}$à la Knuth. Much better than the traditional way to write the Pochhammer symbol:$(x)_n := x^{\overline{n}}$. In a book I'm writing, I use the notation$x^{\uparrow n}$and$x^{\downarrow n}$, which I find much less clumsy (consider$(2x+1)^{\overline{6k-2}}$vs$(2x+1)^{\uparrow6k-2}$). Anyway, the utility in either of these notations is seen in the umbral calculus; it makes the connection to "ordinary" calculus much more apparent, such as with $$\Delta x^{\uparrow n} = n x^{\uparrow n-1}\qquad\text{compared to}\qquad D x^n = nx^{n-1}.$$ 2. The simple idea of omitting parentheses for function application:$f\,x$as opposed to$f(x)$. I think this often makes some mathematics look cleaner, especially when the argument isn't especially complex. It also allows for some nice (= convenient) abuse of notation, such as in $$\left[ (-1)^{p - m - n} z \prod_{j = 1}^p \left( z D_z - a_j + 1 \right) - \prod_{j = 1}^q \left( z D_z - b_j \right) \right] G(z) = 0,$$ where$D_z:=d/dz$. Note this equation isn't a product (entirely); upon expansion, we'd have$D_z G(z)$terms. 3. Do fractions count? Imagine having to write $$\sqrt{(x^2 + 2x + 1)\div (5x^3 - 3x^2 + 2x - 7)}$$ instead of $$\sqrt{\frac{x^2 + 2x + 1}{5x^3 - 3x^2 + 2x - 7}}.$$ 4. Big-O notation. Though often abused, this is a much less clumsy way to express boundedness and asymptotics and errors and even lets you begin to do some algebra with them (provided you're careful). I don't think doing such is as obvious when you write it all out manually. 5.$\square(x)$for the square wave,$\triangle(x)$for the triangle wave,$Ш(x)$for the Dirac comb (seriously, see Appel's "Mathematics for Physics and Physicists"). These are more cute than explicitly useful. 6. Notation used with musical isomorphisms as a way to do raising and lowering of indices. We have$X^\sharp$which raises the index (in the context of Einstein summation) and$X^\flat$which lowers the index. Here,$\flat$and$\sharp$are isomorphisms between tangent$TM$and cotangent bundles$T^*M$:$\flat:TM\to T^*M$and$\sharp:T^*M\to TM$. 7. Using$\operatorname{cis}\theta = \cos\theta + \mathrm{i}\sin\theta$, mathrm{i}\sin\theta$ (cosine i sine), which is nice for obvious reasons (yes, $\omega = e^{\mathrm{i}\theta}$ is nice too) and $\operatorname{cas}\theta = \cos\theta + \sin\theta$, sin\theta$(cosine and sine), which is used in e.g., the Hartley transform. 8. Notations for hypergeometric functions $${}_pF_q \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right) = {}_pF_q(\mathbf{a},\mathbf{b};z)$$ and Meijer-$G$functions: $$G_{p,q}^{m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right)=G_{p,q}^{m,n} \!\left( \left. \begin{matrix} \mathbf{a} \\ \mathbf{b} \end{matrix} \; \right| \, z \right)$$ 9. Notation for general continued fractions: $$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$$ The$\mathrm{K}\$ comes from German's "Kettenbruch", which is "continued fraction."