You did not tell us how large the sample is. If the transformations have failed increase the sample size to see whether according to the central limit theorem the data can follow an approximate normal distribution.

Even then you cannot predict Y with certainty but still under probability. Statistical inference is necessarily uncertain. 

Dear Sindhuraj

If you have just two continuous variables with a non-normal distribution and you want to predict Y values on the basis of X (or vice versa) then you are talking about the regression model. The regression model does not require normal data distributions, but only a normal distribution of residuals. This is a common mistake. The question should not be: do I have a normal distribution of data (because it is not needed at all) only - what is the relationship between variables - linear or not.

If you are working on SPSS, you check dependency by estimating the curves in the regression module. If in Gretle I recommend the RESET test. If you have questions - let me know.

P.S. Transposition of variables is a bad idea, there are many reasons, but the basic one is: there is no way to understand the results. If you have really very bimodal distribution I would recommend rather dividing a variable into eg quartiles and doing a variance analysis / H test (ANOVA or Kruskal)

There is a quite a bit of literature on the advantages and disadvantages of transformation.  Many methods are available, and the specifics of your goals will play a large role here. Without further details, I'll give a very brief summary here:

1) transform your data before running basic linear model (or hypothesis test; which is basically a linear model with factors). The method has been commonly used, but it's important to note that the inference about E[log(y)], where y is your response variable and E[] denotes the expected value, is not the same as the inference about E[x]. So one must be careful about calculating confidence intervals and interpreting the results.

2) Generalized linear models (GLM) are an effective way to deal with data that does not have constant variance and usually have some limits on the values of the dependent variable, y, may take. Unlike the transformations above, GLM models log(E[x]) (log being one of many transformations available as a link function in GLM scheme). Thus, GLM has some advantageous properties over simple transformations.

3) if you data is really skewed, you can use mixture distribution (hurdle, zero-inflated, or tweedy/gamma) models. These are a bit more complicated, but the basic idea is that data comes from a complex distribution usually generated by more than one process. Thus the researcher aims to identify the level of contribution from each process (each process is usually equal to a distribution) as well as the parameters of the distribution.

When it comes to uncertainty propagation, you should look up bootstrap techniques and/or the delta method for uncertainty propagation.

First, check for normality. If it fails, try log transformation and check again. If this also fails, you need a nonparametric test, depending on the case.

Try this paper

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

Normality tests can be tricky as they detect any deviation from normality. And as Damasz pointed out above, normality of data itself isn't essential for regression. More specific complications are: 1) they theoretically rely on a known distribution with known parameters and test for similarity to that distribution; and you usually don't know the "true" parameters. 2) in practice, they are very sample-size dependent. If your sample size is large enough they will usually find deviations for normality, or any other target distribution.

Important note tho is that while overall normality isn't strictly required for OLS, OLS or Gaussian-based likelihood will be sensitivity to non-normal residual distributions. So if you have high degree of skew in data, and especially if that skew is somehow related to the independent variable, OLS estimators won't be consistent, best, unbiased etc... That is the reason for GLM, and mixture-distribution models.

From the question, it seems you have two sets of variables (x & y) and neither of them is normally distributed. Well if they are large in sizes, say both been more than 60, then the central limits theorem (clt) still says its safe to assume normality for both. You did not explain the relationship existing between them, do they occur in pairs? are they covariates resulting from the same experiment or is one an auxiliary or even hidden (latent) variable of the other? Have you tried to measure the correlation between them? depending on the relationship existing between them, the appropriate methods of analyses changes. Do they occur in pairs? For instance a simple linear regression could be used to model them (i.e. obtain y=ax +b type of model) more so if they are positively or negatively correlated or the method of least squares or even by plotting x against y (x,y) in a rectangular coordinate system and finding the line or even curve of best fit etc.