All numbers in JavaScript are actually IEEE-754 compliant floating-point doubles. These have a 53-bit mantissa which should mean that any integer value with a magnitude of approximately 9 quadrillion or less -- more specifically, 9,007,199,254,740,991 -- will be represented accurately.

NOTICE: in 2018 main browsers and NodeJS are working also with the new Javascript's primitive-type, BigInt, solving the problems with integer value magnitude.

Boolean(expression) will simply convert the expression into a boolean primitive value, while new Boolean(expression) will create a wrapper object around the converted boolean value.

The difference can be seen with this:

Copy// Note I'm using strict-equals
new Boolean("true") === true; // false
Boolean("true") === true; // true

And also with this (thanks @hobbs):

Copytypeof new Boolean("true"); // "object"
typeof Boolean("true"); // "boolean"

Note: While the wrapper object will get converted to the primitive automatically when necessary (and vice versa), there is only one case I can think of where you would want to use new Boolean, or any of the other wrappers for primitives - if you want to attach properties to a single value. E.g:

Copyvar b = new Boolean(true);
b.relatedMessage = "this should be true initially";
alert(b.relatedMessage); // will work

var b = true;
b.relatedMessage = "this should be true initially";
alert(b.relatedMessage); // undefined

JavaScript has two number types: Number and BigInt.

The most frequently-used number type, Number, is a 64-bit floating point IEEE 754 number.

The largest exact integral value of this type is Number.MAX_SAFE_INTEGER, which is:

• 2⁵³-1, or
• +/- 9,007,199,254,740,991, or
• nine quadrillion seven trillion one hundred ninety-nine billion two hundred fifty-four million seven hundred forty thousand nine hundred ninety-one

To put this in perspective: one quadrillion bytes is a petabyte (or one thousand terabytes).

"Safe" in this context refers to the ability to represent integers exactly and to correctly compare them.

From the spec:

Note that all the positive and negative integers whose magnitude is no greater than 2⁵³ are representable in the Number type (indeed, the integer 0 has two representations, +0 and -0).

To safely use integers larger than this, you need to use BigInt, which has no upper bound.

Note that the bitwise operators and shift operators operate on 32-bit integers, so in that case, the max safe integer is 2³¹-1, or 2,147,483,647.

const log = console.log
var x = 9007199254740992
var y = -x
log(x == x + 1) // true !
log(y == y - 1) // also true !

// Arithmetic operators work, but bitwise/shifts only operate on int32:
log(x / 2)      // 4503599627370496
log(x >> 1)     // 0
log(x | 1)      // 1

Technical note on the subject of the number 9,007,199,254,740,992: There is an exact IEEE-754 representation of this value, and you can assign and read this value from a variable, so for very carefully chosen applications in the domain of integers less than or equal to this value, you could treat this as a maximum value.

In the general case, you must treat this IEEE-754 value as inexact, because it is ambiguous whether it is encoding the logical value 9,007,199,254,740,992 or 9,007,199,254,740,993.

2nd October 2020: note that many bare-bones approaches are fraught with subtle bugs (eg. whitespace, implicit partial parsing, radix, coercion of arrays etc.) that many of the answers here fail to take into account. The following implementation might work for you, but note that it does not cater for number separators other than the decimal point ".":

function isNumeric(str) {
  if (typeof str != "string") return false // we only process strings!  
  return !isNaN(str) && // use type coercion to parse the _entirety_ of the string (`parseFloat` alone does not do this)...
         !isNaN(parseFloat(str)) // ...and ensure strings of whitespace fail
}

To check if a variable (including a string) is a number, check if it is not a number:

This works regardless of whether the variable content is a string or number.

isNaN(num)         // returns true if the variable does NOT contain a valid number

Examples

isNaN(123)         // false
isNaN('123')       // false
isNaN('1e10000')   // false (This translates to Infinity, which is a number)
isNaN('foo')       // true
isNaN('10px')      // true
isNaN('')          // false
isNaN(' ')         // false
isNaN(false)       // false

Of course, you can negate this if you need to. For example, to implement the IsNumeric example you gave:

function isNumeric(num){
  return !isNaN(num)
}

To convert a string containing a number into a number:

Only works if the string only contains numeric characters, else it returns NaN.

+num               // returns the numeric value of the string, or NaN 
                   // if the string isn't purely numeric characters

Examples

+'12'              // 12
+'12.'             // 12
+'12..'            // NaN
+'.12'             // 0.12
+'..12'            // NaN
+'foo'             // NaN
+'12px'            // NaN

To convert a string loosely to a number

Useful for converting '12px' to 12, for example:

parseInt(num)      // extracts a numeric value from the 
                   // start of the string, or NaN.

Examples

parseInt('12')     // 12
parseInt('aaa')    // NaN
parseInt('12px')   // 12
parseInt('foo2')   // NaN      These last three may
parseInt('12a5')   // 12       be different from what
parseInt('0x10')   // 16       you expected to see.

Floats

Bear in mind that, unlike +num, parseInt (as the name suggests) will convert a float into an integer by chopping off everything following the decimal point (if you want to use parseInt() because of this behaviour, you're probably better off using another method instead):

+'12.345'          // 12.345
parseInt(12.345)   // 12
parseInt('12.345') // 12

Empty strings

Empty strings may be a little counter-intuitive. +num converts empty strings or strings with spaces to zero, and isNaN() assumes the same:

+''                // 0
+'   '             // 0
isNaN('')          // false
isNaN('   ')       // false

But parseInt() does not agree:

parseInt('')       // NaN
parseInt('   ')    // NaN